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Let
denote the subgroup of G generated by P; the elements of
are the products
of points.
32
F. Bachmann
LEMMA 2: (Geom. Dedicata 4, 1975).
a,b
ula:r's if and only if ab
E
are aonneatible by a ahain of perpendia-
.
Hje1ms1ev's AKL contains a variety of surprising lemmas on double incidences.
Some of them can be proved for perpendicularity groups and some others
have a core which is valid in all perpendicularity groups. Let a,b,a,d be a chain of perpendiculars with vertices A,B,C and Ola,d; Hjelmslev says: If
b'
give two examples.
b' :";"""":---:7"""::::::=::::,.., B
is any line joining
A,B, then the fourth reflection line d'
of the concurrent lines d,a,(O,b') is incident with c.
Q
(Fig. 2.)
If a point C and a line u are given,
o
we can draw the perpendicular (C,u) and
c
d'
erect in C the perpendicular on (C,ul.
Fi g. 2
This is the line C(C,u), and it is called the C-pal'alZel of u.
I see the core of Hjelmslev's lemma in a more general lemma
3 which can be nicely proved for perpendicularity groups: LEMMA 3: 0
E
F(ABe) and A,Blu imply OIC(C,u) ,
"For points
A,B,C
holds: The c-para1lels of all lines joining
dent with any fixed point of the product of the reflections in We can say more: Under the assumption 0 bijective map of the set of lines jOining
A,B
A,B
are inci-
A,B,C".
E F{ABC), u --~ C(C,u)
is a
onto the set of lines joining o,C.
Lemma 3 and this extension are often useful, e.g. for the study of neighbour relations. Another lemma of Hjelmslev's says:
Fig. 3 a
If two lines a,b have one and only one
c
point in common, then the perpendiculars dropped from a point C to G,b have only 1
the point e in common.
b
(Fig. 3.)
lOriginal version (AKL, 1. Mitt. 62): If a,b have several points in common, then the perpendiculars dropped from a point C to a,b have several points in common.
Orthogonality in reflection geometry
33
This lemma does not hold in the group planes of Galileian groups.
But we
can replace the assumption that a,b have exactly one point in common by the stronger condition that the product of the reflections in a and b leaves exactly one point fixed, and we can prove for all perpendicularity groups: LEMMA 4: IF(ab)
I = 1 implies
F((C,a) (C,b))
= {C}.
As an application, I sketch a proof of LEMMA 5: (KnUppe1 1980).
IF(ABC)
I:::: 1,
"A product of three point reflections has at most one fixed point".
PROOF: (for perpendicularity groups). Then 0
E
Take a line f through o.
ABCf can be reduced to a product Dg of a point D and a line g
F(ABCf).
,;
(HG 2 .6), and 0
Let 0 E F(ABC).
E
F(Dg) implies that '-
Dg is a line a through 0 (HG 2*.12). b
Thus we get ABC = af
with Ola,f.
d Q
By lemma 2 (and some additional
e
reasoning), it follows that there are lines b,c,d,e such that a,b,c,d,e,f
o
rl Q
Fig. 4
is a chain of perpendiculars and b,e have a point Q in common. cd E P implies IF(cd)
then F(af)
Since a point reflection has exactly one fixed point,
I g 1; we apply lemma 4 twice and get at first
F(be) = {Q},
= {O}.
(Fig. 4), For a translation, induced by a product AB, we see from Lemma 5 that the mapping P - - + P , X ~ midpoint of
x,xA B is
injective; this is needed for
introducing "semi-translations". 5.
We should remember that the group generated by all affine line reflections of the real affine plane contains euclidean, Minkowskian and Ga1i1eian subgroups with the same point reflections.
We may ask if a perpendicularity group (G,S),
with paint set P, contains always Hje1mslev subgroups with the same point set P.
F. Bachmann
34
This is the case.
If for a
E
S, we denote by S((a)) the set of all lines which
are connectible with a by chains of perpendiculars, then «P> a Hjelmslev subgroup of (C,S) and its point set is still P.
U
a
,S(( )) is
(In the subgroup, any
three concurrent lines have a fourth reflection line.) On the other hand, we can construct perpendicularity groups as extensions of Hjelmslev groups. 1974.
Such a construction theorem has been proved by M. Gerth in
Fig. 5 shows the group plane of Gerth's minimal example of a perpendicu-
larity group which is not a Hjelmslev group (In this example, we have
lsi
= 24,
ipl
= 9)1.
Ici = 216,
The perpendicularity group in question is constructed as a
semidirect product of the euclidean group over CF(3) and a cyclic group of order 3; the group plane contains a subplane which is isomorphic to the euclidean plane
over CF(3).
a
Fig. 5
1 Every line of the group plane is incident with three points. line a and three analogous lines consist of two pieces.
In Fig. 5, the
Orthogonality in reflection geometry
35
B1 BLIOGRAPHY 1.
2. 3. 4. 5. 6. 7.
8. 9.
F. Bachmann, Aufbau der Geometrie aus dem Spiege1ungsbegriff (AGS). 2. Auf1. Springer 1973. - Der H8hensatz in der Geometrie invoZutorischer GruppeneZemente. Can. J. Math. 19, 895-903 (1967). - HjeZmsZevGruppen (HG). Mathematisches Seminar, Universitat Kie1 1970/71, 2. Neudruck 1976. - Lotketten in HjeZmslevgruppen. Geom. Dedicata 4, 139-158 (1975). M. Gerth, FoZgerungen aus dem Axiomensystem des Senkrechtstehens. Dip10marbeit Kie1 1974. M. Gotzky, Uber Bewegungsgruppen affin-orthogonaZer Ebenen. Symposium: Neuere Ergebnisse Uber Projektivitatengruppen, Bad Windsheim Ju1i 1980. J. Hje1ms1ev, Einleitung in die Allgemeine Kongruenzlehre (AKL). Danske Vid. Se1sk., mat.-fys. r·ledd. 8, Nr.ll (1929); 10, Nr.1 (1929); 19, Nr.12 (1942); 22, Nr.6, Nr. 13 (1945); 25, Nr.10 (1949). B. K1otzek, Ebene aquiaffine SpiegeZungsgeometrie. ~lath. Nachr. 55, 89-131 (1973). F. KnUppe1, Homomorphismen metrischer HjeZmslevebenen, welche Geraden, die sich eindeutig schneiden, auf ebensolche abbilden. Eingereicht a1s Habi1itationsschrift Kie1 1980. R. Schnabel, 1981a, Kennzeichnungen eukZidischer Ebenen als affiner Ebenen mit SpiegeZungsoperator. Erscheint in Mitt. Math. Ges. Hamburg. 1981b, Uber die Geometrisierung gewisser Permutationsgruppen. r~anuskri pt. Y Why Tschen, AZgebraisation of pZane absoZute geometry. Am. J. r~ath. 67, 363-388 (1945). O. Veblen and J.W. Young, Projective Geometry. Boston I 1910, II 1918.
BUlow-Str. 16 D 2300 Kie1 F. R. Germany
This Page Intentionally Left Blank
Annals of Discrete Mathematics 18 (1983) 37-54 © North-Holland Publishing Company
37
ON Sm·1E TRANSLATION PLANES ADMITTING A FROBENIUS GROUP OF COLLINEATIONS Claudio Bartolone
In this note we state some results concerning with translation planes of dimension 2 over GF(q), where q
= pro From now on IT will denote such a plane.
Assume that IT has a collineation group F of order q2(q_l) satisfying the following cond it i on: there exis ts a point V E as a Frobenius group on £",-{V}.
9.", such that
F fixes V and acts (fai thfu Zly)
This situation actually occurs: (a) let TIl be a
translation plane over a semi field Q of dimension 2 over its middle nucleus Nand suppose that N - GF(q). a group U of
If a
pe~spectivities
E
Q and a
E
N, a f. 0, then (x,y) .... (ax,ax+y) defines
of order q2(q_1).
Clearly the pair (TI 1,U) satisfies our hypotheses; (b) let TI2 be a LUneburg plane [5] of order q2. IT2 possesses a
collineation group S -- Sz (q). If V E £00' it is well known that Sv has order q2(q_l) and operates as a Frobenius group (with non-commutative Frobenius kernel) 2
on £",-{V}; (c) letIT 3 be a Betten plane (see [1] for definition) of order q , q = 2r (r odd) (a Betten plane of odd characteristic is better known as Walker or Betten-Walker plane [8]).
IT3 has an abelian group L of col1ineations which fixes
the line «0,0,1,0), (0,0,0,1» lines through 0
and is sharply transitive on the remaining q2
= (0,0,0,0). L is determined by the following matrix group 0 a
0
0
0
0
b a 3 a +ab b
a
0
0
0
c
0
0
0
0
0 0 2 c 0 3 0 c
/ a,b
E
GF(q)
0
The matrix group 0 / c
E
GF( q)
";(
C. Bartolone
38
defines a co11ineation group C which fixes (only) the lines through 0 < (1,0,0,0), (0,1,0,0) > and «0,0,1,0), (0,0,0,1) > and normalizes L.
Hence LC is a Frobenius
group of order q2(q_1) acting on ioo as our hypotheses require.
(b) and (c) fill
up the class of proper (i.e. no semifie1d planes) translation planes of characteristic 2 satisfying our hypotheses.
In fact we prove (section 3) the following
theorem 1 which classifies these planes (see [2), [4), [7) for further theorems characterizing the LUneburg planes). THEOREM 1: plane.
~et
TI be of characteristic 2
and suppose that
TI is not a semifieLd
If K denotes "he Frobenius kerneL of F, then one of the following holds: (i) K is commutative and TI is a Betten pZane;
(ii) K is not commutative and TI is a LUneburg pZane. When trying to prove theorem 2 in case of characteristic
F 2, one observes
that an analogous conclusion to (ii) cannot occur: in fact a Frobenius group with even order complement has always a commutative Frobenius kernel.
Thus one could
think that the only proper translation planes of odd characteristic satisfying our hypotheses are the Betten-Walker planes.
Nevertheless this is not the case in
that the following theorem holds (see lemma 3): THEOREM 2: If the chm'acteri:stic of TI is F 2, then TI is a semifield plane. Look at the full group of collineations of a Betten-Walker plane TIw of order q2, q
= pr
= -1
(mod 6) (see (8)).
TI
w has a co11ineation group Gw in the linear
translation complement which fixes a point at infinity and acts transitively on the remaining q2 points.
G has as subgroups two p-groups Sl and S2 of order q w one of which, say Sl' is not a shear group. Gw is the split extension of SlxS 2 by a group N of order q-l, yet Gw is not a Frobenius group. However SlN is a Frobenius group with respect to N. Now G characterizes TI in the sense that we w w are specifing: assume that Hand N are two col1ineation groups of the translation plane n such that: (i) H is a p-group of order q2 and N is a group of order q-l normalizing H; (ii) G = HN fixes a point V E ioo and acts transitively on £oo-{V}; (iii) G is not a Frobenius group, nevertheless H contains a subgroup K such that: K~ G and KN is a Frobenius group with respect to N. prove (section 4):
Under these hypotheses we
On some translation planes admitting a Frobenius group
:rHEORHl 3: K has order q, moreover, if and
1T
1T
39
is not a semifield plane, q;: -1 (mod 6)
is a Betten-Walker plane.
1. BACKGROUND INFORMATION (A). plane
According to J. Andre' we can identify the points of a translation
of dimension two over GF(q) with the vectors of V (GF(q)) (vector space of 4 dimension 4 over GF(q)) and assume as 1ines of 1T through 0 = (0,0,0,0) the compo1T
nents of a spread S (a class of 2-dimensional subspaces which form a partition of the non-zero vectors of V4 (GF(q)). The group C of collineations of 1T fixing 0 is called the translation compZement of 1T: each collineation of C acts on the points of
as a semilinear map of V (GF(q)) which sends components of S in components of 4 S. Vice-versa such a semilinear map, by acting on the vectors of V4 (GF(q)), induces a collineation of C. The subgroup L(C) of collineations of C which are in1T
duced by linear maps is named the linear translation complement. suppose that A maps the point (x.) of
1T
1
on the point
(x~); 1
then the action of A 4
can be represented by a 4x4-matrix (a .. ) over GF(q), where x! lJ
(i = 1, ••• ,4).
Let A E L(C) and
1
= J=.L l
a .. x. lJ
J
Later on we will identify A with the matrix (a .. ). A collineation lJ
cr of L(C) is called a shear of axis each line parallel a semifield plane.
to~.
~
when cr fixes each point of the line
If the group of shears of axis
~
~
and
2
has order q , then
1T
is
Further information about translation planes can be found in
[ 6).
(B). We will use the following symbolism for a group G: oG
= the order of G;
ZiG) a the center of G; NG(U) = the normalizer (in G) of U. Let F be a finite group and U a proper subgroup of F.
F is called a
Frobenius group with respect to U when g-lUg n U = 1 for any g E F - U.
The ele-
ments of F which do not lie in any subgroup conjugate to U, together with 1, form a characteristic Hall subgroup K of F (the Frobenius kernel of F) and F is the split extension of K by U. Moreover oU divides !F:U!-l and if N 4 F, then N ~ K Let g E F - K; the map a ~ ag = g-lag induces a fixed-;oint free
or K ~ N.
automorphism (shortly f.p.f. automorphism) of K.
l Therefore a ~ aga- is a
bijection of K onto K and so K is inclosed in the derived group F' of F.
The
center Z(U) of U is not trivial and if oU is even, then U contains exactly one l involution j: j acts on K sending k E K to k- (whence K is commutative). A transitive permutation group is named a Frobenius group when it is a Frobenius
40
C. Bartolone
group with respect to its stabilizer, provided this is not trivial.
Hore informa-
tion about Frobenius groups and other equivalent definitions can be found in [31. In order to semplify the exposition we fix the following HYPOTHESIS A: Let p be a p-g~oup
tension of a
subg~oup
exists a lvspeet
;;0
Let (oK,oN)
p~ime
H of
G be a
and
Ol'de~ p
2r
by a
g~oup.
g~oup
Assume that G is the split ex-
N of
o~de~ p
r
-1.
K of H n01'mal -in G such that KN is a Frobenius
F~the~mo~e g~oup
there
with
N.
Rbe
the Frobenius kernel of KN.
= 1. As K is normal, K~ K or K ~
is a divisor of oK-l (1.1) K "as
Assume K
=
Rhas the same order of R. In any case R= K.
K, say pS, since The order of N
Ps -1, i.e. r divides s whence
o~del" pr o~ p2r
and it is the Frobenius kemel of KN.
H: if H is of exponent> p, then
(1.2) His cf erponent p2; moreove~ H has exactly pr_l elements of o~de~ p e:w/: of iJhie!) iJelongs to
Z(H) and is a
p-powe~
of some element in H.
PROOF: The Frobenius kernel of HN is H by (1.1), hence H admits a group of f.p.f. automorphisms of order pr_ l • As Z(H) I 1 (H is a p-group), Z(H) contains at least r
p -1 elements of order p.
\ =l
Owing to an hypothesis there exist
A,~
E
H such that
= 1: the previous arguments show that H has pr_l elements satisfying the same conditions of A. Now if s E Z(H) and sP = 1, then ('!1;)p = l when \J E H. Therefore for each p-element A E H which is a p-power of I 1 and :\p
some element in H, there are at least pr elements in H having p-power equal to A. An easy computation allows to deduce that there are in H pr (p r -1) elements of order p2 and pr -1 p-elements.
2. PRELIMINARY LEMMAS Let
TI
be a translation plane of dimension two over GF(p r ) I GF(2) and let S
be the spread which defines of
n
TI.
In this section G will denote a collineation group
in the translation complement satisfying hypothesis A of section 1.
In addi-
On some translation planes admitting a Frobenius group
41
tion we claim that G fixes a component V of S and operates transitively on the .. p2r componen t s. If U is a subgroup of G, we will denote by L(U} and remalnlng E(U}, respectively, the group of collineations of U lying in the linear translation complement and the set of shears contained in U.
The subgroup of col linea-
tions of U which fix the set X of points (lines) of n is indicated by Ux ' First of all we will state the following LH1MA 1: H acts reguZarZy on S-{V} and N fixes a further component
U of
S.
2r r PROOF: Since G operates transitively on S-{V}, oG = oG/p = p -1 where E E S-{V1 E 2r Hence from oH = p it follows H () G = 1 and so H acts regularly on S-{V}. E Moreover G = HG whence GEis a complement of H in G as well as N. As H is an Hall E subgroup of G, from a well known theorem of Zassenhaus (see, for instance, B. Huppert [31, pag. 128, theor. 18.3) it follows that Nand GE are conjugated groups in G. Therefore N = GU for a suitable component U of S. Concerning the admissible orders for K we will prove now the following lemmas 2 and 3. LH1MA 2: Let p = 2.
Then K = H or r(K}
K.
PROOF: Assume that K F H: then we see that oK = 2r by (l.l). of K.
Let
j
be a 2-element
By making use of a well known theorem of Baer (see, for instance. H.
LUneburg [61, pag. 20) and of lemma 1. we infer that j is a shear of axis V. OE(K} ~ 2r. i.e. E(K} order 2r _l. LEMr~A 3: Let p F 2.
= K. because N induces in K a f.p.f. automorphism group of
Then oK
PROOF: Assume that oK
= pr or E(K}
K = H.
F pro then (l.l) entails K = H. The order of N is pr_ l •
i.e. an even number: so N contains exactly one involution j. Moreover if .
Thus
~
E
K.
-1
~J
= ~ • By lemma 1 j fixes (besides V = V n ~oo) U = U E ~oo. Let j fix an other point at infinity WF V. then j E GU n GW' As GWis conjugate to N = G by lemma U
1.
is the only involution also for G ' Therefore, if x E K is the collineation W such that UX = W(see lemma l). it results jX = j. But we have pointed out before j that x = x- l : this means x = x- l • that is x = 1 since K is a p-group and p F 2. j
42
C. Bartolone
Hence U = Wand j is an homology of center V or of axis t
j
= x (resp.
1; E
K, ~-
pj
1
.
other words
~
by Baer's theorem.
to V (resp. point P of V).
i
Let
If
must act on the parallel lines to V (resp. points of V) as
= f;J
2
= P) for each parallel line
V
f;.
In
= 1 on the parallel lines to V (resp. points of V). As 2 does not
divide the order of
1;,
we deduce that each element of K is a perspectivity of
center V (resp. axis V), indeed, a shear because lemma 1 holds. REMARK 1: With a proper choice of the base we may assume that U = {(x,y,O,O)/x,y E
GF(pr)} and V
= {(O,O,x,y)/x,y E
GF(pr)}.
Thus if
y E
L(G),
where A,C.V are 2x2-matrices over GF(p r ) and 0 is the zero matrix: in particular 4
implies also C = 0 by lemma 1. As H is a p-group. if xi
9
j~l
hi / j (i = 1, •.•• 4) are the equations which define a collineation of H (regarded as a
'I E
L(N)
semi1inear map of V4 (GF(pr))). it is not restrictive to assume that h .. lJ i < j and, if 8 = 1, h .. = 1 for i = j.
= 0 for
lJ
The map (a ij ) ~ a (resp. (a ij ) ~ a43 ) defines an homomorphism 21 "'2) between L(H) and GF(pr)(+). Now we prove LEMMA 4: either
'p. 1
~(H)
?f~.
1
= 1.2).
(i
kern~l
(resp.
kern~2)'
(resp.
Mopeover K is contained in L(H) and
induces an homomoy?hism of K onto GF(pr)(+)
PROOF: Let x E axis V).
is the kernel
~1
OP
J:(K) = K.
then x is a perspectivity of center V (resp.
Therefore x E J:(H) by lemma 1. Conversely wE J:(H) implies VW = V and
VW = V because lemma 1 holds: hence
w E kern~ .• 1
By (1.1) K is the Frobenius kernel of KN: hence K is a subgroup of the derived group of KN and consequently K is contained in L(H). p2r by (1.1). Assume oK
= pr
K has order pr or
and oJ:(K) > 1. As N induces in K a f.p.f.
automorphism group of order pr_ l , we infer that oJ:(K) ~ pr and consequently K = Z(K).
Therefore under the hypothesis oK a pr and J:(K) F K, ~. is an iso1
morphism of K onto GF(pr)(+). Let oK = p2r; if J:(K) F K. then p = 2 by lemma 3 r and oJ:(K) = 2 by (1.2) (K = H is of exponent> 2 in view of Baer's theorem and lemma 1). This means that kern~. has order 2r whence ~. is an homomorphism of K 1
1
On some translation planes admitting a Frobenius group
REMARK 2: If L(H)
1 r(H)
(this is the case supposing r(K)
43
F K),
by lemma 4 there 0 exists (h .. ) E L(H) such that h21 101 h43 • Hence, if x~ = .il g .. x. are the 1 J= lJ J lJ equations which define a collineation of G, it is g .. = 0 for i < j because lJ vG = V and L(H) ~ G. L(N) acts completely reducible on the points of TI (since
t oL(N»: hence in case L(H) 1 r(H) we can select a base of V4(GF(pr» (in accordance with remark 1) so that the collineations of L(N) are represented by p
diagonal matrices. Lemmas 3 and 4 show us that in case p 1 2 and E(K) 1 K, E(K) is trivial. Nevertheless E(H) 1 1 in any case: in fact LEMMA 5: E(H) is aZways non-triviaZ.
Furthermore each shear of K is an eZement of
the center of H.
PROOF: The first claim is true if oK
a
p2r (lemma 4).
Let E(K) 1 K 1 H, then p 12
and oK = pr (lemmas 2 and 3): in this case E(K) = 1 as we noted above.
As
Z(H) n K 1 1 because H is a p-group and K 4 H, if we make use again of the pr_l f.p.f. automorphisms induced by N in K, we notice that K is contained in the center of H.
Therefore for each element k E GF(pr), there exists (z .. ) E Z(H) 4 lJ with z12 = k (lemma 4). Let I; E H and x~ = .E l h. j. be the equations that define 1 J= lJ J 1;: then h .. = 0 for i < j (see remark 1). By imposing the identity 1;1; = 1;1; for lJ each 1; E Z(H) we find0 = 1. Therefore L(H) = H and we can consider a collineation (a i ) E L(H) - K. By lemma 4 it is possible choose (k ij ) in K with k21 =-~l and so we find the shear (a .. )(k .. ). lJ lJ Assume E(K) = K: then in any case E(K) is contained in Z(H) (for H is an elementary abelian p-group if E(K) = K = H, while we have shown above that E(K) = K ~ Z(H) in case oK = prj. otherwise, E(K) = 1.
Let E(K) 1 K: then we may assume p = 2 because,
By lemma 2 so K = H and by (1.2) E(K)
~
Z(H) since ExpH> 2
(the 2-elements of H are shears). Evidently r(H)
~
H whence Z(H) contains some non-identical shear (s .. ) by lJ
lemma 5. Assume L(H) 1 E(H), then there exists (h .. ) E L(H) with h21 1 0 1 h43 . lJ -1 by l:~ma 4. Thus (sij)(hij)=(hij)(sij) Ylelds s32 = 0 and s31 = s42h21h43' Hence h21h43 has constant value, say k, when (h ij ) runs in L(H) - E(H).
44
C. Bartolone
REMARK 3: Let {e ,e ,e ,e } be the base used to give the coordinates to TI in l 2 3 4 accordance with remarks 1 and 2. The previous arguments show that, by replacing (if that is the case) e l by k-le • we can represent each collineation (h ij ) of l L(H) in such a way that h21 = h43 • Moreover under the hypothesis L(H) ~ I(H) (for example supposing I(K) ~ K) we can set 5 21 = s32 = s43 = 0 and 5 31 = s42 for each shear (s .. ) of the center of H (for examp 1e for the shears of K (lemma 5)). lJ
There-
31 (resp. (sij) ~ 5 42 ) defines a injective map of I(H) n Z(H) into GF(pr) since, otherwise, there would exist a shear (5 .. ) # 1 such that U(sij)nuF1. fore (sij)
~ 5
lJ
LEMMA 6: If I(K) t- K, then k32 aut,cmol"pi7.isn: of Gf(pr). a~so
=
k~l f02' any (kij)E K, UJhel'e a is an additive
Under the hypotheses L(H) = H, I(H) .;;; Z(H) and oE(H)=pr,
the remaining elements of H satisfy the above condition.
In case K is
alJt:liall, xC< = hx ].Jhe1'6 h is a non-zero element of GF(pr).
PROOF: By (1.1) either K = H or oK = pro
Let K = H, then L(H)
=
H by lemma 4.
# K entails ExpH > 2 and, by applying (1.2), we find I(H) .;;; Z(H) and
L(K)
o~(H)
By lerrrna 4 we can consider (a ij ) and (b ij ) in H such that a = -b • 2l 21 From (c ij ) = (aij)(b ij ) it follows c 2l = c 43 = 0 and c = a 32 +b : hence (c ij ) is 32 32 a shear of H. On the ground of remark 3 we may assume so c32 = 0 whence =
pro
= -b 32 • As a consequence a 21 = a43 ~ a32 defines a map a of GF(pr) into itsaf 32 (see lemma 4). Now by a simple calculation one can realize the additivity of a.
a
Suppose that there exists (h ij ) in H such that h21 t- 0 = h32 = h~l. Then, if (5 ij ) is the shear of H such that 5 42 = -h 42 (see remark 3), we find that the nonidentical collineation (h!.) = (h .. )(5 .. ) of H must fix U: a contradiction (lemma lJ
1).
This proves that If K ~ H,
a
lJ
lJ
is an automorphism of GF(pr)(+).
a
is a map and K is an abelian group because it is isomorphic to
GF(pr)(+) (see lemma 4).
Thus if (a .. ) and (b .. ) are collineations of K, from lJ
lJ
(aij)(b ij ) = (bij)(a ij ) we can infer a~lb21 = a21b~1. 50 for b21 = 1 we find a~l = ha , where h = lao Assume h = 0, then it is easy to check that a ~ a 21 21 31 defines an homomorphism B of GF(pr)(+). If there exists a non-zero element k in
ker~. then the components of 5 U and u(k ij ) must be equal in case (k .. ) is the lJ collineation of K with k21 = k. As this contradicts lemma 1, we have proved that f'
is an automorphism of GF({) under the hypothesis h
consider a shear (s .. ) 1J
k31
~
=
O.
Now by 1ell11la 5 we can
1 of the center of H and the collineation (k .. ) of K with 1J
= -5 31 • As 5 21 = 5 32 = 5 43 = 0 (see remark 3) and k32 = 0, the components of
4S
On some transla tion planes admitting a Frobenius group
Thus (k .. ) should be a shear, but E(K) = 1 in lJ this case (lemma 4): a contradiction. Therefore h must be-a non-zero element of
S
U
and U(sij )(kij) must coincide.
GF(pr) and consequently a is an additive automorphism. a To finish the proof, if K is abelian and K = H, one can prove that x = hx in the same way as in case K F H (clearly h F 0 because we have proved already that a is an automorphism of GF(pr)(+)). REMARK 4: If E(K) F K, h
=
la is a non-zero element of GF(pr). -1
Refer to the base -1
of remark 3: if we replace the vectors e l and e 2 by h el and h e2 , then we note a a that, for any (k ij ) E K, k32 = k21 = k43 where a is an automorphism of GF(pr)(+) such that la = 1. Thus a is the identity map in case K is an abelian group. We shall now denote our attention to the collineations of N.
First of all
we prove: LEl4t4A 7: Under the hypothesis E(K) F KJ eaah aolZineation v of N aan be represented by equations x~
1
= c.x.e(i 1 1
1, ••• ,4; c. F 0). 1
bijeation 1ji of N onto GF(pr);, by setting 1ji(v) =
Moreover one defines a
c c -l 2 l
PROOF: First we point out that oL(N) > 1 because oL(N) ~ oN/oAutGF(pr) = pr-l/r>l: thus there exists a non identical collineation
y
= (c lJ .. ) in L(N) (according to
remark 2 we may set c .. = 0 for i F j). Let (k .. ) E K - E(K) (whence lJ lJ k21 F 0 F k43 ), then (kij) = (kij)Y is again a collineation of K - E(K) (different from (k .. ) because Y induces a f.p.f. automorphism in K). lJ k c c -1 k' k' k -1. 21 22 ll = 21 = 43 = 43 c44 c33 ,l.e.
This yields
(2.1 ) If P F 2 (whence oK = pr by lemma 3), then k21 F k21 by lemma 4. again if p = 2 (and K = H): for if k21
= k21 ,
This is true
then by lemma 4 there should exist a
shear a in K such that (k!.) = (k . . )a. Since a E Z(H) (lemma 5) we have 2 Y2 lJ 2 lJ 2 ((k .. ) )Y = ((k .. ) ) = (k!.) = (k .. )a(k . . )a = (k .. ) • Thus Y should fix the nonlJ lJ lJ 2 lJ lJ lJ -1 identical collineation (k ij ) of K. Therefore from k21 k21c22cll it follows (2.2) If x~
1
4
E d .. xe. are the equations that define the collineation = j=l lJ J
v
of N, we
C. Bart%ne
46
may set d .. = 0 for i < j (see remark 2). N
lJ
U = U and L(N)
i >
j.
~
Besides, by imposing the conditions
N and by making use of (2.2), we notice that d .. = 0 even if lJ
Now if we set (k':.) lJ
=
(k .. ) v, we can prove, as we did above for lJ
y,
that
k21 1 k21 = k~ld22d11 -1. Thus the map v ~ k~ld22d11-l must be injective when k21 is a fixed non-zero element of GF(pr). By setting k21 1, we see that V ~
d d 22 l1
-1
r ,',
is a bijection of N onto GF(p)
r :',
since oN = oGF(p ) .
8: Under the hypothesis E(K) F K, the map Wof lemma 7 sends L(N)onto a sub-
lEt~
group D of GF(pr)*oontaining the multiplioative group of a field
GF(pn) 1 GF(2).
PROOF: In the proof of lemma 7 have pointed out that OL(N) F 1: thus one can verify easily that D = IjJ(L(N)) is a non-trivial subgroup of GF(pr/'. If we set r * =and o(N/L(N)) = m, we see that D =. If m = 1 there is nothing GF(p) to prove: so we may assume m> 1.
The automorphisms of GF(pr) used to represent
the col1ineations of N form a cyclic group A (isomorphic to N/L(N)). As oA = m, e pn the automorphi sm x .... x = x , where mn = r, generates A. let v E N be inherent to
then the coset \JL(tl) generates the factor group N/L(N).
=),
the collineation of N such that 1jJ(\1)
Therefore if
\1
is
= a, there exists a positive integer t such
n 2n (1:-1) n s q l+p +p + ••• +p q that \lE (vL(N)). If IjJ(v) = a , then we find that a = (am) (a ) for a. suitable positive integer s: hence ms+q(l+p n+p 2n +•.. +(t-1)n )=1 (mod pr -1).
t
As m divides pr_ l , we deduce (q,m) n
(a q)l+p +... +p n
k
(m-l)n
= a1+p +•.• +p
-_ (am)
z
= 1. Now from v m E
L(N) it follows
: so m must be a d"lVlsor 0 f 1+p n+••. +p (m-1)n. , l.e.
(m-l) n E
O.
But k generates the multiplicative group of GF(pn),
the fi xed fi e 1d of 0 in GF( pr).
Therefore we must prove only that e does not
generate AutGF(pr) in case p = 2.
let ~
E
Z(N), ~ F 1.
If ~ is inherent to an
automorphism a F 1, a would fix each element of D. Consequently r would have a proper divisor u such that 00 / 2u_1 and we should find the contradiction 2r_1 = = oN = oDo(N/L(N)) .;; (2 u-1)r < (2 u_l)(2 u(r/u-l)+2 u(r/u-2)+ •.. +2 u+l) = 2r _1. Hence a
= 1 and
that
j
1jJ(r,;)
E
D.
Now (W(r.))0 = w(~) 1 1 follows from r
does not generate AutGF(p ).
vI,; = r,;v
and so we note
47
On some translation planes admitting a Frobenius group
3. PROOF OF THEOREM 1 We may assume that F is contained in the translation complement C of
TI
since
F is isomorphic to a subgroup of C satisfying the same conditions of F (for each collineation of
TI
is the product of a collineation of C and of a translation and,
besides, F does not contain translations). section 1 and is transitive on too-{V}.
Clearly F verifies the hypothesis A of
Therefore we can make use of results and
In our case K = Hand E(K) f K because
symbolism of section 2.
is not a semi-
TI
field plane. K is of exponent> 2 since the 2-elements of K are shears in view of Baer's theorem and 1emma 1. Hence each shear (s .. ) of K is a square by (1. 2): (s .. ) = 2 lJ lJ = (k ij ) for a suitable collineation (k ij ) of K. Thus we see that s31 = k21k32 r r and consequently k21 ~ k21k32 is a 1 - 1 map of GF(2 ) onto itself (for OE(K) = 2 by (1.2) and (s .. ) lJ
~
s .. is an injective map as we have pointed out at the end of lJ
remark 3). Therefore the map (3.1)
a
of lemma 6 satisfies the condition
x ~ xx a is a 1 - 1 mapofGF(2 r ) onto itself
Let (k ij ) and (kij) be two co11ineations of K such that k21 = k21 f O. On the ground of remark 3and lemma 6we have also the identities k43=k21=k21 =k'43 and k32 = k32 • The co11ineation (kij) = (kij)(kij) is a shear (because k21 = 0) of the center of K (see lemma 5): according to remark 3 so k31 = k42 and consequently k31 + k31
k42 + k42 • Hence k21
k31 = k42 + k~l for such that 08 = O. (3.2)
any
~
(k ij )
k31 + k42 is a well defined map 8 and we find E
r K, where 8 is a map of GF(2 ) into itself
Let (k ij ) and (kij) be two co11ineations of K such that k21 = k21 and k42 = k42 (so k31 = k31 , k32 = k32 , k43 = k43 )· As (k ij ) and (kij) act in the same way on 23r points of TI, we infer (k .. ) = (k! .). This means that each lJ
lJ
co11ineation (k .. ) of K is different from the remaining col1ineations owing to the lJ
value of the coefficients k21 and k42 • Therefore~: (k 21 ,k42 ) bijection of GF(2 r )xGF(2 r ) onto K and it is easy to check that (3.3)
~ (x,y)q>(x'
,y' )
cp( x+x' ,y+y' +xx' a).
~
(k ij ) is a
48
C. Bartolone
The coefficient k41 of (k ij ) E K must be necessarily a function of k21 and k42 : k41 = f(k ,k 42 )· Thus from ~(x,y)~(O,y) = ~(x,O) it follows f(x,O) 21 = f(x,y)+xy+f(O,y) and, by setting f(x,O) = xY and f(O,y) = yO, we find 6 f(x,y) = xi +xy+y.
By imposing the identity
~(x,O)~(y,O)
=
~(x+y,xy
eL
) now we
deduce Y
, tfle
0
k41 = k21k42+k21+k42 for any (k. j ) E K, where Y and ':1 l ' en;:;z. t is then isomorphic to l 2 3 123 L • For any X in E, since D(E) == {-A, 0, + A}, there exist A, 11, v in lF3 3 such that: for instance e
'IT
(e"e 2 ,X) = AA,(e"e 3,X) = IlA and (e 2,e 3,X) = vA. By trilinearity the element u = vel -
lle
2
+ Ae 3 satisfies also:
(e l ,e 2 ,u) = AA, (e ,e 3,u) = IlA and (e ,e ,u) = vA. 2 3 l Hence X' = X-u is orthogonal to t' in
'IT.
So there is no
~oss
'IT,
of
in the sense that (t,t',X')
genera~ity
0 for any t and
in assuming that e , eS, ••• e are
n 4 In case n = 4 one gets e E Z(E) so that E ~ IL3 x F • When 4 3 n a S, consider the kernel K of the homomorphism x -~ (x,e 4,e ) from 'IT into D. S If K = 'IT, then e and e are central elements and necessarily E ';; IL3 x If 4 S
orthogona~
to
'IT.
IF;.
K F 'IT, then K is a maximal subloop of
and we can choose e 2 and e3 in K. By changing eventually e in -e 4 , one may assume then that 4 (e l , e4 , eS) = A = (e l , e 2, e3 ). We are thus led to R ';; liS where liS is the 3-CM ~oop on five generators e"e ,e ,e ,e subjeat to (e"e ,e ) = (e"e ,e ) and. for 2 3 4 S 2 3 4 S 'IT
any i < j < k suah that {l,4,S} F {i,j,k} 'f {l,2,3}, (e.,e.,e ) = O.
A direct k study of liS shows that K(ITS) = 2 and IITsl = 3. Besides Z(IT S) = D(II S)' which means that lIS may not be written as a direct product of smaller 3-CM loops (lIS is 6
"irreduaible").
1
J
We have thereby proved the:
THEOREM 2: There exist
exaat~y
two non isomorphia
5-dimensiona~
HTSs of size 6.
This completes the proof of the fact that there exaatly four pair-wise non-
60
L. Beneteau
isomorphic
HTS~
of size 6, since the unicity in the 4-dimensional case was
established in 11 J.
The next step would be the classification of the 6-dimen-
sional HTSs of size 7, which is to be dealt with in the same way, though the list of the corresponding systems is doubtless much longer.
BIBLIOGRAPHY 1. 2. 3. 4. 5. 6. 7. B.
9. 10.
L. Beneteau, Contribution d l'etude des Boucles de Moufang commutatives et des espaces apparentes, (Algebre, Combinatoire, Geometrie), These d'Etat, Univ. Marseille, 1981. L. Beneteau, Topics about Moufang Loops and Hall triple Systems, "Simon Stevin", vol. 54, n02, pp. 107-124, Avril 1980. L. Beneteau, Problemes de majorations dans les quasigroupes distributifs et Zes gl'oupes de Fischer, Actes ColI oque "A 1gebre app 1i quee et combinatoire", pp. 22-34, Univ. Sc. et Med. de Grenoble, Juin 1978. L. Beneteau, Free commutative Moufang loops and anticommutative graded rings, Journal of Algebra, Vol. 67, nOl, pp. 1-37, November 1980. L. Beneteau, Une classe particuliere de matrotdes parfaits, Proc. Colloque franco-canadien de Combinatoire, Juin 1979; Annals of discrete Math. 8, pp. 229-232, 1980. R.H. Bruck, Contribution to the theory of loops, Trans. Amer. Math. Soc. 60, pp. 245-354, 1946; t1R 8 # 134. M. Deza, Finite commutative Moufang loops, related matroids, and association schemes, to appear in Proc. Conference on Combinatorics, 1979, Arcata, California, Humboldt State Univ., Utilitus t·1ath. J. I. Hall, Note on the orders of Hall triple systems, to appear in Journ. Comb. Th. Ser. A. Marshall Hall Jr, Autcmorphisms of Steiner triple systems, 1Bt4 J. Res. Develop., pp. 460-472, 1960; MR 23 A = 1282. Yu. 1. Mani n, Cubic forms; North Ho llandp. C., Amsterdam, 1974.
Universite Paul Sabatier, Math., 118, route de Narbonne 31077 Toulouse Cedex, France
Annals of Discrete Mathematics 18 (1983) 61-76 © North-Holland Publishing Company
61
ON MAPPINGS PRESERVING A SINGLE LORENTZ-MINKOWSKI-DISTANCE. I Walter Benz
Consider a commutative field K and denote by K2 the set of all ordered pairs of elements of K.
The elements of K2 are called points of the plane K2.
The
distance PQ of the points P 1:1 (PI' P2)' Q = (ql ,q2) is defined by PQ:=
(ql-pt}(q.-P.).
The following result was proved by F. Rado, [3]:
THEOREM: Asswne char K + 2,3. "P,Q
E
K2
Given a bijection a
K2 ... K2 such that
PO = 1 if and only if pOQo = 1.
Then a must be semi linear up to a translation (considering
~
as a vector space
over K).
In case K 1:1 JR we have proved in
[1]
the sharper result
THEOREM: GiVen a mapping a of the plane JR2 into itself such that
Then a must be semi linear up to a translation and hence a Lorentz transformation.
The main question we are interested in this series of three papers I, II, I II is
PROBLEM A: Determine all mappings a of K2 into itself such that
62
W. Benz
We shall present the solution of this problem for infinitely many fields K, especially for infinitely many Galois fields by applying a result of G. Tallini, An important step in our considerations is the following generalization (for
141.
char K 4 5,7) of Rado's theorem: THEOREM 1:
Ae2wne
"P,Q ~"h;!n
E
chell" K + 2,3,5,7.
Given an injective mapping
0
K'
-->
K'
such
K'
a "lust D'] l3emiZiiwar (with respect to a monomorphism of K and not necessari-
Zy an
a, 3.
induction.
I{x 1"
Case A : We assume
Now apply
.. , xn} I > 2.
+ X, without loss of genera 1 ity.
Xl
Put P.1'- ( Vfi Xv' v~i l). X
Then, by induction,
v
p. ° = (vfi 1
,
Ii'
:J (u,v) p.
l).
\"
'Jfi
'J
Put P .-
(jJ
(LX ,L v
l) and Xv
pO
=: (u,v).
So flP '
1
1 implies
Ii
1, i. e.
1
(u -
Z
(j)) (v -
i:
v+i
v+i"
\i
1. v
Hence v
(2 )
= 1 and i = 2 we get
Applying (2) for (u-L4J) u .-;:",
(4'2
0 impl ies v =
v
U - Lr;J
"
+ O.
( lVl
+
(U-IID)
L:
q:tl)
(u - L W
)
V
We have
(j)l
u - L
i.e. u =
;;3
So (2) (put v = -
1
""
f
4'2
- either 5 n T = 0 or 5 n T E F;
II. V f E F, 3! 5 E S, 3! T E T : f
~
5, f
~
T;
III. If 5, 5', 5" E S pairwise meet at distinct points, then any element in S
other than 5" meeting 5 and 5' at distinct points meets 5". A3 • There exist a line f E F and aTE T, with f ~ T, and a complete, finite chain of subspaces of (G,F) such that its length is h+l, its minimum is f and its maximum is
T.
Let (G,F) and (G',F') be two projective Grassmann spaces having finite index
hand h', respectively.
Then the collections Sand T (S' and T') of
maximal subs paces in (G,F) ((G',F')) are defined.
Any mapping F : G --~ G' which
is one-to-one and onto will be called an isomorphism between (G,F) and (G' ,F') if the following hold: (a) F' is the set of all images under F of the elements in F',
s' is the set of all images under F of the elements in S; (y) T' is the set of all images under F of the elements in T. (s)
In this paper the following result will be proved (see sect. 8).
THEOREM: If (G,F) is a connected projective Grassmann space of finite index h, then a projective space IP exists and (G,F) is isomorphic to the h-th Grassmann space (Gh(IP), Fh(!p)) associated with !P.
116
A. Bichara and G. Tallini
2. SOME PROPERTIES OF A PROJECTIVE
SPACE HAVING A FINITE INDEX
GRASS~~NN
h ;;. 2. Let (G,F) be a connected PPLS satisfying axioms A , A2, and A3 in sect. 1. 1 Then: PROPOSITION I: ::Ohe cdlections Sand T ape skew. i.1.1G
Let
t~
and W (with t·l
tl') be
~
ma:l.'ima: subspacec having two distinct common points p and p'; then M and M'
belong t:o d.iffe-pent coUections of maximal subspaces.
Thus, two distinct maximal
aubsracC!s }xdonging to the same coUection have at most one common point.
PROOF: I f a max i ma 1 subspace tr' were conta i ned inS nT, then M" n rl"
W; a
contradiction to A I, as f.1" properly contains a line. 2 Since p and p' belong to t~, they are collinear; let f be the line joining them; obviously, f
~
Mand f
~
tl', the two maximal subspaces belong to different
collections and the statement is proved. PROPOSITION II: :"et Sand S be tUJO distinct m=imal subspaces in S, having a I
If T is a maxirrul subspace in T, meeting Sand S' at the lines f
common ?oint p.
and f', respectively, then f and f' meet at p, so T passes through p.
PROOF: f and f' are distinct lines (otherwise, two distinct maximal subs paces S and S' in S would meet at the line f = f', which is impossible by prop. I). any line in F has at least two points, there exist points p' that p, p' and p. are pairwise distinct. (indeed, p, p'
E
S', and p', p"
f and p"
E
f' such
These points are pairwise collinear
T); thus, by A , there exists a sub1 space N through them and N is contained in a maximal subspace M. If M belonged to E
S, p, p"
E
Since
E
S, then 11 = S (by prop. I, H n S' :: {p,p'} implies
(since 11 n S' :: {p,p"}).
~1
= S); similarly,
Therefore, if 11 belonged to S, then S
=
I~
=
S'
S' a contradic-
tion.
Hence '·1 f/. S and, by A , 11 E 1. Since t~ belongs to T and contains the 2 distinct points p' and p" in T E T, by prop. I,ll = T. Moreover, p E 11 and M T imply pET.
The point p in T belongs to S; thus, {p}
~
Tn S
=
f and p E f;
similarly, p E f'; therefore, the distinct lines f and f' meet at the point p and the statement is proved.
117
On a characterization of Grassmann spaces
PROPOSITION III: Any Tin T is a projective space. PROOF: It is enough to prove that in the line space T Veblen-Wedderburn axiom holds: Let fl and f2 be two lines in T meeting at the points P3; if f3 and f4 are distinct lines in T, each of them meeting both fl and f2 at points distinct from P3' then f3 and f4 meet at a point. Set {Pl}
=
f2 n f3' {P2}
=
fl n f3' {ql}
= fl
n f4' {q2}
f2 n f 4; then
(2.1) Through fi (i = 1, ••. ,4) there is exactly one maximal subspaces Sl
S (see A2II). Such maximal subspaces are pairwise distinct (if i 1 j and S. = S., then the J
1
maximal subspace S. would share with T E T the points in f. 1
1
U
E
f., which is J
impossible by A2I). Now, fi C Si' i = 1,2 and {P3} = fl n f2 imply {P3}':: Sl n S2'. since Sl 1 S2' by prop. I, {P3} = Sl n S2' By the same argument, {Pl}· = S2 n S3 and {P2}
= Sl
tively.
The three maximal subspaces Sl' S2' and S3 pairwise meet at distinct
n S3; moreover, S4 meets Sl and S2 at the points ql and q2' respec-
points (see (2.1)) and S4 meets Sl
and S2 at distinct points; therefore, by
A2III, S4 meets S3 at a point q: {q}
= S3
n S4'
lines f3 and f4' respectively, and S3 n S4
Since T meets S3 and S4 at the
= {q}, by prop. II, f3 and f4 meet at
the point q and the statement is proved. The projective space which are members of T, contain projective planes forming a collection
IT
of subsets of
Clearly, any element in
G.
IT
is a subspace
in (G,F) which is contained in a maximal subspace belonging to T. PROPOSITION IV: Let T and T' be two elements in T through a point p in G. S1'
S2~
Assume
and S3 are three pairwise distinct elements in S through p and such that
Tn S.1 = f.1
E
F, and T' n S.1 = f~1
E
F.
same plane a in T, then also the lines
If the lines f., i 1
f~ 1
PROOF: Let a' be the plane in T' through
= 1,2,3,
belong to the
belong to a unique plane in T'.
fi
and
fZ'
It is enough the prove that
f3 belongs to a'. Let f and f' .be two lines not through p, the former in a, the latter in
,
a •
Since a and a' are projective planes, f meets f , f2 and f , and f' meets fl and l 3
118
A. Bichara and G. Tallini
f '. 5et {q.} = f () f., i = 1,2,3, and {q~} = f' () f~, j = 1,2. It is easy to 2 1 1 1 1 check that the five points q., q~ are pairwise distinct; moreover, if 5 and 5' are 1
1
the maxima1 subspaces in S through f and f', respective1y, then the five subspaces 5,5',5 ,5 2,5 are pairwise distinct and {qi } = 5 () Si' {qj} = 5' () 5j" 5ince 51' 1 3 52' and 53 pairwise meet at distinct points and 5' (f S) meets Sl and 52 at distinct points, Sand 5' have a common point q, which is obviously distinct from
q;.
ql and
Thus, the three maximal subspaces S, 51 and 5' pairwise meet at dis-
tinct points and S3 (f S') meets 5 and Sl at the distinct points q3 and p (respectively).
Therefore, S3 and S' have a common point q'.
f3' S' nT' = f' and and f3 are coplanar. thus, it is
(I'
{q'}
Since S3 () T' =
= S3 n S', by prop. II, f' and f3 meet at q.
Hence, f'
The plane through them contains f' and the point p on f3;
and f3 belongs to a'.
3. THE PARTIAL LINE 5PACE (S,R) ASSOCIATED WITH (G,F) Take pEG, a E IT, with pEa; then the subset r
of S, consisting of p,a those maximal subspaces in S meeting a at lines in F through p, i.e. r
p,a
={SES:S3p
and
snaEF},
is uniquely defined. PROP05ITION V: If
a,
a'E
IT
and pEa, p' E a', then
Ir p,a I;;.
(3.1) (3.2)
Ir p,a
PROOF: Since
a
2;
r, ,1;;.2=>r p,a p ,a
r, p
,a
"
is a projective plane, through the point p in
two distinct lines fl and f2 of F.
a
there are at least
The maximal subspace 51 and 52 in S through
fl and f2 (respectively) belong to r p,a and now it will be shown that Sl f 52' If 51 = 52' then this member of S would share with a maximal subspace T in T a set I containing the distinct lines fl and f2' which is impossible by A I.
2
Therefore,
E rp,a and Sl f $2; (3.1) follows. Let $1 and 52 be two distinct elements inr p,a nr, p ,a " SlandS 2 meet at lines in F through p and (I' at lines in F through p'; thus p and p' belong to
Sl' S2
slnsZ'
Byprop.I,p=p'.
119
On a characterization of Grassmann spaces
When a = a
,,
(3.2) is obviously true. Assume a f a'.
Let T and T' be the
maximal subspaces in T through a and a' respectively; then T f T' (otherwise, Sl would meet T at two lines in F through p, one belonging to a, the other to a', which contradicts A2I) and TnT' = {p}. Furthermore, by prop. IV, any element S3 in rp,a' distinct from Sl and S2' meets T' at a line in F, which - being coplanar with Sl nT' and S2 n T' - belongs to r p,a ,; hence, r p,a -C r, p ,a " the same argument, r, (3.2) follows. p ,a ,C - rand p,a
By
Thus, the collection {r p,a : a E n, pEa} of subsets of S is defined; since it is not a proper collection (see prop. V), let R be the proper collection associated with it. PROPOSITION VI: The pair (S,R) is a PLS.
Moreover, two distinct elements in S are
collinear in (S,R) iff they have a common point in G.
PROOF: Let Sand S' be two distinct elements in S.
If S n S' = {p}, then let f be
a line in S through p; through f there is a maximal subspace T E T meeting S' at one point al least; thus, it meets S' at a line f' E F.
In the projective space
T, the distinct lines f and f' are joined by a plane a E IT; therefore, Sand S' belong to r p,a If S n 5' = ~, no element in R through 5 and S' exists. Let f be a line in S
E
S.
Through f there is a maximal subspace T E T.
P E f, let f' be a -line in T through p, distinct from f.
If
Through f' there is a
maximal subspace 5' E S, which is obviously distinct from S and meets S at p; thus, through S there is an element of R (joining 5 and 5'). of S.
Hence, R is a cover
11oreover, for any r E R, j rj ;;;. 2 (see (3.1)) and any two elements in R have
at most one common point (by (3.2)); it follows that (S,R) is a PLS. Let p be a point in G.
The collections S of all elements in S through p is p
a subspace of (S,R).
(Indeed, any two distinct elements in Sp are collinear in (S,R) and the line through them is completely contained in S ). p
PROPOSITION VII: Let p be a point in G and T an element in T through p.
Then the
subspace S in (S,R) is isomorphic to the star F T consisting of the lines in F p p, through p and belonging to the projective space T. Thus, S is a projective space p
and is of finite dimension h iff T is of finite dimension h+l.
120
A. Bichara and G. Tallini
PROOF: Any element in $ meets T at a line in F T (see A I). p p, 2 Let ~ be the mapping defined by :SE$-+snTEF T p,
4'
-1
is one-to-one and onto (see A2II). Moreover, ~ maps pencils onto lines, in R, belonging to $. (Indeed, any pencil in F T consists
Clearly, in F
p, T
~
P
p,
of all lines in T through p, belonging to a plane a E IT; such a pencil is the image under
of the line r p,a ). Now, it will be proved that
r
P.!l
Q
maps lines in $p onto pencils in F r Let p, , be a line in S and Sand S' two distinct points on it; they meet T at ~
P
(distinct) lines f and f', which are joined by a plane a through p in T; then, r p,a ,'" r p,a (see prop. V); furthermore, fjJ(r p,a ) '" q>(r p,a ,) is the pencil consisting of the lines in Fp, T through p and belonging to a. It follows that
Q
is an isomorphism between $p and the star of lines through p in the projective space T. From prop. VII, it follows immediately: PROPOSITION VIII: Le" T and T' be any two distinct elements in T through a point p in G.
Then T is of finite dimension h+l iff T' is of finite dimension h+l.
Next, we prove: PROPOSITION IX: Any T in T is an (h+ 1 )-dimensional projective space. the coZlection $
p
Furthermore,
oj all elements in $ through a point p in G is an h-dimensional
projective space, which is a subspace of ($ ,R).
PROOF: If T is the space Assume
T in
T, T f
I;
T in
A , then T is an (h+l)-dimensional projective space. 3 let ql be a point in f and q2 a point in T. Since (G,F) ;s
connected, there exist both a finite subset {Pl, .•• ,Pn} of pOints in G and a finite subset {fl, .•• ,fn_ll of fi
~
lin~s
{Pi' Pi+ll, i = 1. 2, .... n-1.
in F such that Pl
= ql'
Pn
= q2
and
Through any line fi there is exactly one Ti
in T (see A2II). The subspaces T and Tl have the common point Pl '" ql; since has finite dimension equal to h+l, by prop. VIII, Tl is an (h+l)-dimensional projective space.
T.1 n T.1+ 1
::l {po
-
1+
1) (i '" 1, ... ,n-2) and T 1 n T ::l {p }, all nn
T
On a characterization of Grassmann spaces
T.'s (i 1
= l, ••• ,n-l)
121
and Tare (h+l)-dimensional projective spaces.
Finally, to prove that Sp is an h-dimensional projective space, it is enough to recall that there exists an element T in T through p (through p there is at least one line f in F which is contained in a maximal subspace in T); T is an (h+l)-dimensional projective space and from prop. VII the statement follows.
4. THE SUBSPACES OF (S,R) Now, we prove the following: PROPOSITION X: Let Sl' S2' S3 be any three pairwise collinear in (S,R) points of S, and meeting at a same point p of G, as subspaces of (G,F); then there exists a projective plane in (S,R) through them.
PROOF: The family Sp consisting of all elements in S through p turns out to be a projective space of dimension h
~
2 (see proposition IX) in (S,R), which contains
Sl' S2 and S3' PROPOSITION XI: Let Sl' S2' S3 be any three pairwise collinear in (S,R) and independent points of S, which do not meet at a same point, as subspaces of (G,FJ; then a projective plane in (S,R) through them exists.
PROOF: Set Pl
= S2
S3 ' P2
n
= Sl
n
S3 ' P3
= Sl
n
S2
(and such points exist); they turn out to be pairwise distinct (otherwise, if Pl = P2 then Pl E S2 ' Pl E Sl' that is P3 = Pl = Sl n S2' and thus Pl = P2 = P3)· Then there exist in F three lines (P l ' P2), (P l ' P3), (p 2' P3), which are pairwise distinct (otherwise, Sl' S2 and S3 contain the line (P l ' P2) = (P l ' P3) and this cannot occur because of A ,II). Hence, a subspace of (G,F) (see A ) containing 2 l Pl ' P2 and P3 exists. Such a subspace, say a, belongs to IT: a meets Sl at a line, hence it is contained in a maximal subspace T of T and turns out to be the projective plane a
= {S
E
a
joining Pl ' P2 and P3 in T.
S : S n'a
E
F}
Next, we prove that
122
A. Bichara and G. Tallini
is a projective plane.
Let S, and S2 be any two distinct elements of
meet the projective plane
a
~':
a ;
they
at two distinct lines fl and f2 of F, respectively,
which intersect in a point p of
Thus, Sl' and S2' belong to the line r in R. ,., p,a p') be any two (distinct) lines in a . Since the line a.
Let rand r, ,(p tp,a p ,a (p,p') in F on ex exists, the maximal subspace S E S through it is the only element belonging to r
p,a
n
r, ,. p ,a
The statement follows.
As a corollary to prop. X and XI: PROPOSITION XII:
Ar,y subspace of (S,R) is a projective space.
5. THE COLLECTION P OF MAXH1AL SUBSPACES IN (S,R)
PROPOSITION XIII: Let; p be a point in G and S an element of S through q (c G) and
q not th!'ougiz p fsince (G,F) is a PPLS, such a subspace exists).
Then, the set
P ad S n S t- 0 } is either a Zine in R 01' the empty set. By q proF. IX, the t3Zements in S :;h!'ough p [m'm an h-dimensional (h ;;. 2) projective L = {S
E
S: S
::J
spaoe in (S,R); tin.s, there is always some element in S through p which is skew with Sq; consequently, (S,R) is a propel' partial space (see VI).
PROOF: Assume that, through p, there are two distinct elements of S, say S' and S", both meeting Sq at a point: S' n Sq q', S" n Sq = q". Then, q' t- q". (Indeed, q' = q" implies that through the line (p,q') (p,q") there are two distinct elements of S, a contradiction).
Since the three points, p, q', and q" are pairwise
collinear, there exists a subspace
in
in (S,R) p,a consists of those elements in S through p, meeting S at a point on (q',q"); a
IT
through them.
The line r
q
moreover, any point on (q' ,q") is joined to p by a line belonging to some element in r
p,a Now, assume S'" is an element in S through p, neither skew with S , nor q
belonging to r p,a ; then, S"' meets Sq at a point q"' not on (q',q"). Since the points 4,
A.
Since
has infinitely many lines and only
rr
line L which is not a secant of A. and there can be at most A.
(~) secants
of
A,-rr
rr
has a k-arc
certainly has a
Each secant of A meets l in a unique point,
(~)distinct
points of l lying on one or more secants of
Since l has infinitely many points, there certainly exists a point· P of l
lying on no secant of A. (k+l)-arc of
In particular, P is not in A.
Clearly Au {P} is a
This proves Lemma 2.1.
rr.
By the remark in the second paragraph of this section, applied with k = 6, a projective plane containing a 6-arc must have order at least 4.
This condition
is also sufficient, as follows from Lemma 2.1 and the next lemma: LEMMA 2.2: (i)
be a projective plane of finite order n, 0here n > 4.
~et rr
Then:
quadrangle of rr is contained in precisely (n-2)(n-3) distinct 5-arcs
Ev~ry
of rr. (ii) Every 5-arc of rr is contained in precisely 1 + (n-4)(n-5) distinct 6-arc of rr.
REMARK: In view of Lemma 2.2, the numbers of 5-arcs and 6-arcs in a projective plane of order n can be given as explicit polynomials in n. The proof of Lemma 2.2 is very simple and need not concern us here.
We
wish, instead, to point out that 5-arcs and 6-arcs have played important roles in the theory of Pascalian projective planes (i.e., projective planes coordinatized by fields.) k-arc.
Such a plane has conics.
A set of k distinct points of a conic is a
Every 5-arc lies in exactly one conic.
Pascal's theorem on conics, taken
with its converse, characterizes the 6-arcs (call them Pascalian 6-arcs) which lie in a conic, and the "constructive" theory of conics is essentially the theory of Pascalian 6-arcs. There are also non-Pascal ian 6-arcs, and these have received little attention. Now let us focus our attention on a 6-arc S of a projective plane (About
rr
we assume only that its order is at least 4, possibily infinite.)
is a pOint of
rr
which is not in S, then, for some integer i
i secants and 6-2i tangents of S. points of
rr.
rr
Clearly
°
~
i
~
3.
~
0, P lies on exactly
Thus S partitions the
which are not in S into four disjoint subsets
If P
141
Enriched cliques with six vertices
where, for 0 < i < 3, S. is the set of all points of 1
act1y i secants of S.
rr
not in S which lie on ex-
(Some of these sets may be empty.)
We may regard S as a clique on 6 vertices, which we denote by 1,2,3,4,5,6, and we may identify each edge (ij), i line of
rr
~
j, of the clique with the unique secant
containing the points i,j of S.
say (12), (34).
As lines of
rr,
Now consider two disjoint edges of S,
these must meet in a unique point P of
not in S and hence must be in S2 or S3'
rr
which is
If P is in S2' we denote it by (12) (34).
If P is in S3' then P must also lie on the unique secant of S disjoint from (12), (34), namely (56); in this case we denote P by (12) (34) (56). Thus each unordered triple, such as
(12), (34), (56)
of 3 mutually disjoint secants of 5,
must either define a unique point, (12) (34) (56), of 53' or three distinct points (12)(34), (12)(56), (34)(56) of 52'
In other words, 53 may be identified with
some (possibility empty) subset of the set of all 15 spreads of our clique, and then 52 may be constructed from the complementary spread-set, each spread of the latter set contributing three points of 53 forming a triangle with the edges of the spread as sides.
Conversely, if we know 52' we also know 53'
Notice that if we denote the cardinal number of 5.1 by f(i): f(i)
=
15il
(2.1)
then f(2), f(3) are non-negative integers related by the equation f(2)
3 { 15 - f(3) }.
(Of course, frO) and f(l) will be infinite unless
(2.2) rr
has finite order.)
Next suppose that we start from our clique 5 on six vertices, choose a subset, S3' of the set of all spreads of S, and define 52 as indicated above in terms of the complementary spread-set.
Then we have a set of
6 + f(2) + f(3)
51 - 2f(3)
142
R.H. Bruck
distinct points, namely 5 u 52 u 53' together with a set of 15 distinct lines (the edges or secants of 5) and a natural criterion for whether or not a given point lies on a given line. The result is a non-degenerate partial plane in the sense of Marshall Han [1] and hence can be imbedded (perhaps only by free adjunction) in at least one prOjective plane.
That is:
LEI1MA 2.3: EVe!'y spI'ead-set of oW' clique on 6 vertices can be identified wi th the 53 of a ,'-arc 5
·~n
at least one (possibly finite) projective plane.
Next suppose that 53 has been chosen (we don't claim to know how) so that the above-described partial plane can be imbedded in at least one finiteprojective plane lines.
We know the point-sets 5, 52' 53 and their relations to secant
rr.
We wish to describe the tangent and non-secant lines in terms of what
points of 5 u 52 u 53 they contain and then define the points of 50 and 51 in terms of these lines and the secants.
Can this be done? The next lemma gives a
partial answer.
LEMMA 2.4: Let
rr
be a projective pLane (of order at least
4, possibily infinite)
which contains no proper pY'ojective subplanes of oY'deY' gY'eateY' than 3.
Then
e,'el'Y t-arc of rr generates rr.
COROLLARY 2.5: Let and n I 16.
rr be a pY'ojective plane of finite oY'deY'
n, wheY'e 4
~
n
~
19
Then every 6-aro of rr geneY'ates rr.
PROOF OF LEMMA 2.4: If S is a 6-arc of w which does not generate IT. then S generates a proper projective subplane, ITO' of
IT.
However, ITO must have order at
least 4. a contradiction. PROOF OF COROLLARY 2.5: By a well-known lemma, if a projective plane of finite 2 order n has a proper subplane of order m, then either (a) n = m or (b) n
~
2
m
+
m.
In view of the inequality n
~
19, (b) cannot occur with m 2 4. and
(a) cannot occur with m ~ 4 unless m = 4, n n = 16. it follows that
rr
=
16.
Since the hypothesis rules out
has no proper subplanes of order greater than 3.
the result follows from Lemma 2.4.
Hence
143
Enriched cliques with six vertices
Finally, let us consider a 6-arc S in a finite projective plane of order n ~ 4 and define the non-negative integers f(i) by (2.1) not only for i = 2,3 but also for i
= 0, 1. Then, in addition to (2.2), we may easily establish the equa-
tions f(O) + f(l) + f(2) + f(3) f(l) + 2f(2) + 3f(3)
= n2 + n - 5 ,
(2.3)
15 (n-1).
(2.4)
We may use these three equations to solve for f(O), f{l) in terms of f(3) as follows: f(O) f(l)
(n - 7)
=
2
+ 6 -
f(3) ,
(2.5)
3 {5(n-7) + f(3) }.
(2.6)
We also notice that 3f(0) If n
+
f(l)
= 3(n - 4) (n - 5).
= 4 or 5, we deduce from (2.7) that f(O)
(2.7)
= O. Thus we get spe-
f(l)
cific values for the f(i) in these cases: If n
= 4, then f{O) = f(l) = f(2) = 0, f(3) = 5.
If n
= 5, than f(O) = f(l)
We ignore the case n
=
= 0, f(2)
= 15, f(3)
6, since there is no plane.
For n
(2.8) 10. =
(2.9)
7,8 or 9 we get the
upper bound 6,7 or 10, respectively, for f(3) (and the lower bound 0.) For n ~ 10, there are no obvious restrictions in f(3) except the inequalities
o :s; f(3)
::;; 15.
In the previous context, a difficult result of Denniston (2) (and an equivalent, point-line dual of this by Thompson et al (3) may be stated as follows:
R.H. Bruck
144
10, then f(3) < 15.
If n
(2.10)
By an elementary use of linear coding, once (2.10) is known, we can improve it to If n
10, then f(3)
~
12.
(2.11 )
-- I will leave (2.11) and other facts about arcs in the (perhaps non-existent) projective plane of order 10 for another paper. Results like (2.10), (2.11) apply to aZZ 6-arcs (in a given projective plane).
We wish to end this section with inequalities applying to some 6-arcs.
First we need two lemmas. LEMMA 2.6: Let Then
1T
7r
be a
has 7-arcs.
7-a:r.Y! of
(possibly infinite) projective plane of order at least ?
Indeed, every 5-arc of rr can be imbedded in at least one
1r.
PROOF: For
1T
of infinite order, the result follows from the proof of Lemma 2.1.
Hence we may assume that The proof for n
1T
has finite order n, where n> 7.
= 7 is difficult. We need the fact that a projective plane
of order 7 is Desarguesian.
From this it follows that every 5-arc can be
imbedded in a conic, which is an 8-arc. The proof for n o
~
i
~
~
8 is quite elementary.
2, let Fi be the set of all points of
secants of F.
Consider a 5-arc F and for 1T,
not in F, lying on exactly
The content of Lemma 2.2 (ii) is that jFol
= 1 + (n-4)(n-5) = n2 - 9n
+ 21
(2.12)
Now suppose that there exist two distinct points P,Q of FO such that the line PQ is a non-secant line of F.
It follows at once that the set F U {P,Q} is a 7-arc
containing F. Finally, suppose that no non-secant line of F contains more than one point of FO'
If P is in FO' than P lies on exactly 5 tangents and n + 1 - 5
n - 4
Enriched cliques with six vertices
non-secants of F.
We conclude from this and (2.12) that there are at least (n - 4) (n
distinct non-secants of F. 5
2
- 9n + 21)
(2.13)
Now we count the non-secants in another way. There are
10 secants of F.
exactly (2)
145
Each point of F lies on exactly 4 secants and
n + 1 - 4 = n - 3 tangents of F; hence there are exactly 5(n - 3) distinct tangents of F.
5n - 15
Since
2
(n + n + 1) - 10 - (5n - 15)
n
2
- 4n + 6
there are exactly n2 - 4n + 6 distinct non-secant lines of F.
(2.14 )
The result of subtracting (2.13) from (2.14) is
6 - (n - 4) { (n
2
- 9n + 21) - n}
2
6 - (n - 4)
(n - lOn + 21)
(n - 4)
(n - 3)(n -7) ,
= 6 -
and this is negative for n ~ 8.
Hence we have a contradiction which proves Lemma
2.6. It follows from Lemma 2.2 that a projective plane of order less than 6 has no 7-arcs.
Since there is no projective plane of order 6, Lemma 2.6 is best
possible. Now let T be a 7-arc of a projective
plane~.
A point P of
will be said
to be in T.1 provided (a) P is not in T and (b) P lies on exactly i distinct secants of T.
Clearly T. is empty for i > 3. 1
We define (2.15 )
146
R.H. Bruck
for i = 2,3.
(Clearly g(O) and g(l) will be infinite if
rr
is infinite.)
We wish
to establish the formula g(2) + 3g(3)
105.
(2.16)
First, however, let us contrast the theory of 7-arcs with that of 6-arcs. Certainly we may regard a 7-arc T of a projective plane" as a clique with 7 vertices which we call 1,2,3,4,5,6,7, and we may identify an edge (ij) i 1 j, of the clique with a secant line of T. (34).
Now consider two disjoint edges of T, say (12),
As secants, these meet in a point P of
rr
which is not in T, and must be in
exactly one of T , T . If P is in T , we may denote it by (12)(34). However, if 2 3 2 P is in T3 , P lies on a third edge, disjoint from (12), (34), and this must be exactly one of the three edges (56), (57), (67), so P may be identified with exactly one of (12)(34)(56),
(12)(34)(57),
(12)(34)(67).
Thus, if we identify T3 with a set of spreads (or unordered triples of mutually disjoint edges of T) there is a condition to be satisfied, namely: Two disjoint edges of T lie
in at most one of the spreads in T . 3
(2.17)
When we bear (2.17) in mind, we see that (2.18) counts in two ways the numbers of unordered pairs of disjoint edges of T. Now we are ready for a lemma. LEMMA 2.7: Let T be a
7-a~c
of a
p~ojective
plane rr.
seven t:-arcs 5 contained in T partition the set T3.
Then the sets 53 for the In particular, there is at
least one 6-arc 5 contained in T for which the corresponding f(3) satisfies
f(3)
~ ~ g(3) S 5 .
(2.18)
PROOF: Let 5 be one of the seven 6-arcs contained in T, and let P be a point of 53'
Then P lies on exactly 3 disjoint edges of S, and these are edges of T.
Hence P is in T . 3
Thus
147
Enriched cliques with six vertices
(2.19) for every 6-arc S contained in T.
Conversely, let P be a point in T3 . Then Plies The 6-vertices of T contained in these edges
on exactly 3 disjoint edges of T.
constitute a unique 6-arc S contained in T, and P is in S3.
Hence the S3's for
the seven 6-arcs S contained in T partition T3 . (We note that some or all of these sets may be empty.) Therefore the average value of f(3), taken for the seven 6-arcs, is precisely g(3)/7. f(3) for at least one S in T.
Thus
:;;:t g(3)
(2.20)
Since g(2), g(3) are non-negative integers, we see from
(2.17) that g(3)
~
35.
(2.21)
Inequalities (2.20), (2.21) prove (2.18) and complete the proof of Lemma 2.7. Let us go a little further for the case that rr has finite order n (with
= 0,1,2,3, are non-negative
n ~ 7.) Then the g{i), defined by (2.15) for i integers.
In addition to (2.16). we have the easily proved equations 2
g(O) + g(l) + g(2) + g(3) = n + n - 6,
(2.22)
g(l) +2g(2) +3g(3) = 21(n-l). From (2.16) and (2.22) we may solve for g(O), g(l), g(2) in terms of g(3) as follows: g(O)
(n-10)2 + 20 - g(3)
g(l)
3 { 7(n-11) + g(3)}
g(2)
(2.23)
= 3 (35 - g(O)).
In particular, if 7 ~ n S 13, then g(3) < 35, and hence (by Lemma 2.7) every 7arc must contain a 6-arc with f(3)
~
4.
Again, if 8
inequalities are improved to g(3) < 28, f(3) become g(3)
~
3.
~
n ~ 12, the corresponding
Finally, if n = 10, they
S 20, f(3) S 2. (There are also lower bounds for g(3), and hence for
148
R.H. Bruck
a suitably chosen f(3), if 7
~
n
~
10.)
It is my belief - as yet, largely untested - that in any projective plane of suitably high (possibly infinite) order every spread-set (of any size from
o to
15) of our clique on 6 vertices, can be realized as the S3 of some 6-arc S The following question concerns a possible plane in which this may not be
of~.
tl'ue. ;~oes
b:g
C:JO
(a)
~ 'JaS
t>,e1'e exiBt
C!
(possibly infinite) projective plane
7r
with the foHOUJ-
vmperti es? 6-aI'CS ani e<Jery 6-arc of
~
~.
can be imbedded in a 7-arc of
(b) 7'hD inequalit!J f(3) 2 5 holds for every 6-arc of
~.
When we compare these properties with Lemma 2.7, we deduce that f(3) = .5 for every 6-arc of
~
but hard to believe.
and g(3) = 35 for every 7-arc
I find this intriguing
of~.
(Note that, for (a), we need only require that
at least 11; the conclusion that g(3)
= 35 would then require that
~
~
have order
have order
at least 14 - and hence at least 15.) 3. E-CLIQUES OF RANK 1 Let rank
p
p
be one of the integers 0,1,2,3.
We shall say that an E-clique has
provided that some edge of the E-c1ique has rank
p
and no edge has higher
rank. In the next section we shall present a decoding algorithm for E-cliques which may be described in terms of a tree with E-c1iquesas nodes.
The root of
the tree is an E-clique which we wish to decode (by finding all spread-sets represented by it, or showing that there are none.) a spread-set attached to it.
Each node except the root has
Each spread-set represented by a node other than
the root (if the node represents a spread-set) is disjoint from the spread-set attached to the node and, in union with the attached spread-set, forms a spreadset represented by the root.
Only the root can have rank 3.
If the tree
branches at a node (which may be the root) then the node has rank 2 and there are precisely three branches.
The nodes along any path from the root have decreasing
size and non-increasing rank; the corresponding attached spread-sets have increasing size and are simply ordered by inclusion. at the end of a branch) can have rank 1.
Only an end-node (or node
An end-node is one of the following:
149
Enriched cliques with six vertices
(a) an E-clique which does not represent a spread-set (according to a simple test); (b) an E-clique which represents a simple spread-set (possibly the empty spread-set) known by means of the algorithm; (c) an E-clique of rank 1.
The
algorithm allows us to write down all spread-sets (if any) represented by the root, provided we can decode the end-nodes of rank 1. The present section is devoted to an analysis of the E-clique of rank 1 with the aid of the theory of 6-arcs in a projective plane of order 4. Let 5 be a 6-arc projective plane n of order 4.
The points of 5 are denoted
by 1,2,3,4,5,6; the secant, or edge, through points i,j of 5 (iFj) is denoted by (ij).
Each point of
* lies
on exactly 5 distinct lines of n, and each line of n
contains exactly 5 distinct points of distinct secants of 5.
n.
Each point of 5 lies on exactly 5
Hence there are no tangents of 5, and the 21 lines of 5
consist of 15 secants and 6 non-secants of 5. points of 5 and 5 points of 53'
The 21 points of 5 consist of 6
Each point of 53 lies on exactly 3 distinct
secants of 5 (disjoint as edges of 5) and is defined by the spread consisting of the 3 disjoint edges.
Each point of 53 also lies on exactly 2 distinct non-
secant lines.
Each secant line contains 2 distinct points of 5 and 3 distinct
points of 53'
Each non-secant line contains no point of 5 and 5 distinct points
of 53' Call two distinct paints of 53 disjoint if they do lie on a common secant line.
Thus they lie on a unique common non-secant line.
Through each of the two
points there is a unique second non-secant line, and the two latter non-secants meet in a third point of 53 disjoint from the first two.
Thus 3 mutually disjoint
points of 53 either lie on a common non-secant line or form the vertices of a triangle with non-secants as sides.
5uch triangles go in "polar" pairs.
Indeed,
let L , L , ... , L6 be the six distinct non-secants in some order, let T be the l 2 triangle with sides Ll , L , L , and let T' be the triangle with sides L , L , L . 2 3 4 5 6 The vertices of T are three mutually disjoint points of 53' and so are the vertices of T'.
Consider the 9 points of 53 which are neither vertices of T nor
vertices of T'.
These 9 points must lie in threes on the three sides of T and
also in threes on the three sides of T'.
Each such point, in fact, must lie on
exactly one side of T and exactly on side of T'. through a vertex of T.
Now consider a secant line
This line contains exactly three points of 53'
these is a vertex of T (the one under consideration.) site side of T and hence on a unique side of T'.
One of
A second lies on the oppo-
The third must be the opposite
150
R.H. Bruck
vertex of T'.
In this way we see that the 9 secants passing in threes through
the 3 vertices of T also pass in threes through the three vertices of T'; the two sets of vertices are "covered" by the same nine edges. that there are exactly
t (~)
=
We note, incidentally,
10 such "polar" pairs.
From the above discussion it should be clear that a set of 4 or 5 distinct, mutually disjoint points of 53 must lie on a common non-secant. Now we have enough to discuss the [-cliques of rank 1 in simple terms.
(We
could also consider all [-cliques in the same vein, but at a high cost in complexity. ) As it turns out, every [-clique of rank 1 is equivalent (under permutation of the vertices of the clique) to exactly one of the [-cliques specified by their incidence arrays as follows (listed in order of descending size): X 1
X0 1
X 0 1 1
X
o X1
oX0 1 oX0 ox
X
X 0 1
o X1
X
X0
1 1 X 1 1 (a)
o1
X0 0
0 1
o X0 o0 X
1 lOX
X 1 100 0
1 X 1 000
X
o1
1 X 000
X 1 0 0
1
(d)
(c)
(b)
X 1 000
o X0 o0 X 1
lOX 0
oX
X
X 0 0 1
0
000
X 1 000 0 1 X 0 000
o0
X 1 0 0
001 X 1 0
000 X 1
001 X 0 0
OOOlXl
000 1 X
o0
000 1 1 X
OOOOlX
o0
0 1 X
(c' )
(d' )
0 0 X 1
(b' )
Notice that if we add the "null" array (the incidence-array of the [-clique of rank 0) and denote it by (a'), the operation of interchanging 0 and 1 induces the permutation (a,a') (b,b') (c,c') (d,d') on the eight arrays and reolaces an array of size k by one of size 5-k for
lSI
Enriched cliques with six vertices
o~
k ~ 5.
A sufficient description of each of the 7 E-cliques follows.
(Orbit
and orbit length refer to spread-sets rather than E-cliques). (a) Size 5.
Represents 6 spread-sets.
One orbit, length 120.
description: A clique (with 6 vertices) on the edges of length 1.
Graphic
Geometric de-
scription: The six spread-sets are the points of the six non-secsnt lines.
(b) Size 4.
Represents 2 spread-sets.
One orbit, length 30.
Graphic de-
scription: The 3 edges of rank 0 form a spread (here the spread (12) (34) (56).) Geometric description: The 2 spread-sets consist of four points other than the
common point (here (12)(34)(56» (c) Size 3.
of one of two distinct non-secant lines.
Represents one spread-set.
One orbit, length 60.
Graphic de-
scription: The 6 edges of rank 0 form a closed cycle of length 6 (here (12), (23),
(34), (45), (56), (61» on the 6 vertices of the clique.
Geometric description:
There exist exactly two spreads all of whose edges have rank 0 (here (12)(34)(56) and (16)(23)(45»
and these are disjoint.
The unique spread-set consists of the
remaining 3 points of the non-secant line through these points. (d) Size 3.
Represents 2 spread-sets.
One orbit, length 10. Graphic de-
scription: The 6 edges of rank 0 partition the 6 vertices of the clique into two
closed cycles of length 3 (here (12), (23), (31) and (45), (56), (64).)
Geomet-
ric description: The 6 paints of the 6-arcare partitioned into 2 triangles (here
1,2,3 and 4,5,6).
The 2 spread-sets consist of the 3 vertices of each of two
"polar" triangles with non-secant sides, covered by the 9 secants containing one vertex of each triangle of the 6-arc. (c') Size 2.
Represents one spread-set.
One orbit, length 60.
Graphic de-
scription: The 6 edges of rank 1 form a closed cycle of length 6 (here (12), (23),
(34), (45), (56), (61»
on the 6 vertices of the clique.
Geometric description:
The spread-set consists of the two spreads (here (12) (34) (56) and (16) (23) (45»
all of whose edges have rank 1. (d') Size 2.
These are disjoint.
Does not represent a spread-set.
No orbit.
Graphic descrip-
tion: The 6 edges of rank 1 partition the 6 vertices of the clique into two closed
152
R.H. Bruck
cycles of length 3 (here (12). (23), (31) and (45). (56), (64)). exists no spread whatever all of whose edges have rank 1.
Hence there
Geometvic descviption:
None. (b') Size 1.
Represents one spread-set.
One orbit, length 15.
Gvaphic
desC1>iption: The 3 edges of rank 1 are mutually di sjoi nt and fonn a spread (here
(12) (34) (56).)
The spread-set consists of this spread.
Geometvic description:
Essentially identical. One final remark.
If the list at the end of the paper were complete
(rather than limited to spread-sets of size at least 8) each of the 6 incidencearrays (a), (b), (c), (d), (c'), (b') would appear (and (d') would not.) Instead, these 6 arrays are represented by arrays equivalent to the complementary arrays and of sizes 10, 11, 12, 13, 14 respectively. 4. DECODING DETAILS In
§
3 we gave a general description of our decoding algorithm in terms of
a tree E-cliques as nodes. decode.
The root of the tree is the E-clique we are trying to
We find all spread-sets represented by the root by examining all end-
nodes of the tree (together with their attached spread-sets) and also decoding the end-nodes of rank 1 with the help of the discussion in
§
3.
In the present section we will give the details for forming the tree. (This is not always unique.) In some cases the tree consists of a single node, serving as both root and end-node.
This will be true of the tree for the E-clique of rank 0 (which re-
presents the empty spread-set) and of the tree of an E-clique of rank 1 (which may represent 0,1,2 or 6 spread-sets according to the case; see
§
3.)
Other
cases will be clear from the discussion which follows. It will be convenient to separate the discussion into two parts, one concerned with E-c1iques of rank 3 and the other with E-c1iques of rank 2. E-CLIQUES OF RANK 3 (Recall the convention that an E-clique of rank 3 occurs in a decoding tree only if it is the root.)
Consider an E-clique of rank 3 and suppose, first, that
we have before us a spread-set of which it is the E-clique.
By definition (See
§
153
Enriched cliques with six vertices
1) an edge of the E-clique has rank i only because the spread-set contains exactly i of the 3 spreads containing that edge.
In particular, an E-clique cannot
have two disjoint edges, one of rank 3 and the other of rank 0, if it is to represent a spread. Thus, if an E-clique of rank 3 has an edge of rank 0 disjoint from an edge of rank 3, the E-clique does not represent a spread.
(This is one case where an
E-clique of rank 3 is the only node of its decoding tree.) Next consider an E-clique of rank 3 with the property that, for every edge of rank 3, each of the 6 edges disjoint from this edge has positive rank. the spread-set
~
as follows: A spread is in
edges of the spread has rank 3.
~
We form
if and only if at least one of the
(It should be clear that a spread-set represented
by the E-clique, if any exists, must contain
~.)
Now let A,S be the incidence
arrays of the given E-clique of rank 3 (the root) and of the E-clique associated with~.
(The latter E-clique also has rank 3.) Consider the difference array ,~
A
=A
- S •
It can occur that A" has a negative off-diagonal entry.
In this case, our
given E-clique (with incidence array A) does not represent a spread, and is the only node of its decoding tree. In the contrary case, A* is the incidence array of an E-clique, and this will be the unique second node of the decoding tree of our original E-clique. thi~
second node we attach the spread-set
~.
To
We denote some obvious properties.
First of all, every edge of our clique which had rank 3 for the root has rank 0 for the second node.
In addition, no edge has greater rank for the second node
than it has for the root.
Finally, the size of the second node is at least 3 less
than the size of the root. Since every spread in
o for
~
contains an edge having rank 3 for the root and rank
the second node, it follows that no spread in
one exists) represented by the second node. disjoint) of
~
~
can be in a spread-set (if
In fact the union (necessarily
and a spread-set represented by the second node will be a spread-
set represented by the root; and every spread-set (if any) represented by the root will have this form. The second node cannot have rank 3 but it might have rank 0 or 1 (in which case it is the unique end-node) or rank 2 (in which case we must examine it
154
R.H. Bruck
further.} E-CLIQUES OF RANK 2 (Such an E-clique may either be the root of a decoding tree which we are just beginning to construct or a node other than the root (but not necessarily adjacent to the root) of a decoding tree in course of construction. latter case, we assume that it has an attached spread-set
~O
In the
with the following
two properties: (a) LO has no spread in common with any spread-set represented by the E-clique; (b) the union of
~O
with any spread-set represented by the E-clique
is a spread-set represented by the root.} Consider an E-clique of rank 2, with incidence-array A.
Choose an edge of
the E-clique of rank 2 and consider the 3 distinct spreads containing this edge. There are three possibilities of interest: (a) At least two of the three spreads contain one or two edges of rank O. (b) Exactly one of the three spreads contains one or two edges of rank O. (c) None of the three spreads contains an edge of rank O. We may explain our interest as follows. before us with the E-clique as its E-clique.
Suppose that we have a spread-set The chosen edge has rank 2 only
because the spread-set contains exactly two of the three spreads containing the edge.
Another edge has rank 0 (if such an edge exists)
only because the spread-
set contains none of the three spreads containing that other edge.
Hence, if the
spread-set exists, we must either have case (b) (in which case we know the two spreads in the spread-set containing the chosen edge of rank 2) or case (c) (in which case there are three possibilities for the two spreads.) If, for the chosen edge of rank 2, case (a) occurs, the E-clique represents no spread and may be taken as an end-node of the decoding tree we are constructing.
If, however, case (b) or (c) occurs, there may be another edge of rank 2
for which case (a) occurs.
For this reason, there may be more than one way of
constructing the decoding-chain. Note however that if the E-clique either has no edge of rank 0 or has at least one edge of rank 0 but no edge of rank 0 disjoint from an edge of rank 2, then (c) is the only possibility, regardless of the choice of edge of rank 2. Again, if the E-clique has an edge of rank 0 disjoint from one or more edges of rank 2, each of those edges of rank 2 must yield case (a) or case (b). may be worth while to examine the edges of rank 0 (if any.)
Hence it
155
Enriched cliques with six vertices
Now let us assume that we have decided to base our construction procedure on a particular edge of rank 2. If case (a) occurs, we take the E-clique as an end-node of the decoding chain. If case (b) occurs, let Il be the unique spread-set consisting of the two spreads with the chosen edge of rank 2 as one edge and with no edges of rank O. Let B be the incidence-array of the E-clique associated with I l , and consider the difference array
(Remember, our given E-clique had A as incidence-array.) This time A": has no negative off-diagonal elements; it is the incidence-array of an E-clique which is the unique successor-node of our E-clique in the decoding chain we are constructing.
To this successor-node we attach a spread-set I, where I = Il if our E-
clique (with incidence matrix A) is the root of the tree, and E-clique is not the root and has attached spread set IO' tively if our E-clique is not the root) that
~
~
= ~O
u
~l
if our
We may verify induc-
has the requisite two properties.
If case (c) occurs, the situation is much the same except there are three different successor-nodes; the decoding tree has three branches at our given E-clique. In either case a successor-node has length 2 less than the given node, at least one less edge of rank 2 and at least one more node of rank 0, and rank at most 2. We shall stop here with the general discussion and consider two specific examples. If our root is given by (1.5) and so has rank 3, then the set
~
(defined in
the discussion of rank 3) consists of every element of the spread-set (1.2) except the spread (14)(25)(36).
The unique successor-node of the root has rank
and size 1 and represents only the l-element spread set consisting of (14)(25) (36).
Hence the decoding tree is a chain with two nodes, and (1.2) is the only
spread-set represented by (1.5). -- In particular, our remarks about (1.7) were unnecessarily elaborate, °a lthough they were appropri ate in the context of the general problem.
156
R.H. Bruck
For our second example we take as root one of the E-cliques covered by the catalogue at the end of the paper, namely the E-clique with incidence-array X 3 1 222
(4.1 )
3 X2 1 2 2 1 2 X3 2 2 213X22
222 2 X 2 22222 X This has rank 3.
Hence our first step is to subtract from (4.1) the array of the
spread-set (4.2)
(12)(34)(56), (12)(35)(46), (12)(36)(45), (15)(26)(34), ( 16 )( 2 5)( 34 ) . Thi s 1eaves
(4.3)
X0 1 2 1
o X2 1 1 1 2 X0 1
2 lOX 1
X1 1 1 X
The array (4.3) has exactly two edges of rank 2, namely (14) and (23).
Hence any
spread-set represented by the array must contain exactly one of the following sets k1' k2' l:3 and also exactly one of the following sets ~4' l:5' ~6:
(14)(23)(56), (14)(25)(36).
l:4: (14)(23)(56), (15)(23)(46).
l:2: (14)(23)(56), (14)(26)(35).
l:5: (14)(23)(56), (16)(23)(45).
(14)(25)(36), (14)(26)(35),
Z6: (15)(23)(46), (16)(23)(45).
~1:
~3:
We wark with the edge (14) and subtract from (4.3) the arrays of Zl' Z2' l:3'
(4.4)
Enriched cliques with six vertices
157
getting, respectively, C = X 0 1 0 1 1, l o X1 1 0 1
C2= X 0 1 0 1 1, o X1 1 1 0
C= X0 1 0 1 1 3 o X2 1 0 0
1 1 X0 1 0
1 1 X0 0 1
12XOOO
o 1 o X1 o 1 1 X0
o 1 oX1 1 o 1 X0 o1 1 oX
o lOX o0 1
1 0 lOX
(4.5)
1 1
X1
001 1 X
At this point in our construction of the decoding tree we have 5 nodes, including the root (4.1), the node (4.3) with associated spread-set (4.2), and for i
= 1,2,3, the node C.1 for which the associated spread-set is the union of (4.2)
and
The nodes C1 , C2 are end-nodes. the fo 11 owi ng; ~i'
Oecoding by the method of
§
3, we find
C1 represents only the set {(13)(26)(45), (15)(23)(46), (16)(24)(35)}. (4.6) C represents only the set {(13)(25)(46), (15)(24)(36), (16)(23)(45)}. 2 Combining with
~1
or
~2
as appropriate, we get two spread-sets represented by
(4.3), namely (13) (26) (45),
(4.7)
(14)(23)(56), (14)(25)(36), ( 15 )( 23 )( 46 ) , (16)(24)(35) ; (13)(25)(46) ,
(4.8)
(14)(23)(56), (14)(26)(35), (15 )(24 )(36), (16)(23)(45) . To get spread-sets represented by (4.1), we must combine each of these with (4.2) - but it is convenient not to do this at present.
Note that (4.7), (4.8) contain
~4' ~5'
respectively. The node represented by C3 is not an end-node, since it has rank 2.
In
158
R.H. Bruck
fact, (23) is the unique edge of rank 2.
Since (14) has rank 0, it follows that
any spread-set represented by C must contain ~6' On subtracting the array of ~6 3 from C we get the array which represents (13)(24)(56). Hence there is a third 3 spread-set represented by (4.3), namely (13)(24)(56) ,
(4.9)
(14)(25)(36), (14)(26}(35), (15)(23)(46), (16)(23) (45). Thus, using a coding-tree with 6 nodes, we have found that (4.1) represents exactly three distinct spread-sets, obtained by combining (4.2) with each of (4.7), (4.8), (4.9). It seems worthwhile to go a little further. E-clique defined by (4.1)
By making a sketch of the
or otherwise, we may verify that the automorphism
group, G, of the E-clique has order 8 and is generated by the permutations (12) (34), (13)(24), (46).
Obviously G maps the spread-set (4.2) upon itself.
addition G maps (4.9) upon itself.
In
Hence the union of (4.2), (4.9) is a spread-
set represented by (4.1) with stability group G, orbit length 720/8 = 90.
The
permutation (12)(34) fixes each of (4.7), (4.8), but each of (13)(24), (56) interchanges (4.7), (4.8).
Hence (4.7), (4.8) are equivalent and each has stabil-
ity group H (in G) of order 4 generated by (12}(34), (13)(24)(56).
Thus the
union of (4.2), (4.7) (or of (4.2), (4.8)) is a spread-set represented by (4.1) with stability group H, orbit length 720/4 a 180. At this point we have given c complete discussion - aside from the matter of degrees and anti-degrees - of entries numbered 31, 32 in the catalogue at the end of the paper. Although this second example was worth examining, it is far from typical. Consider, for example, the 21 orbits of size 9 listed in the catalogue. Sixteen of these are represented by E-cliques which represent a unique spread-set.
Two
others (numbered 44, 50) are each represented by an E-clique which represents two equivalent spread-sets.
Another (#52) is represented by an E-clique which repre-
sents 4 equivalent spread-sets.
Finally, orbits numbered 53, 54 are represented by
the same E-clique and this, in turn, represents a total of 5 spread-sets,two belonging tooneorbitand three to the other. A similar situation holds for sizes 10 and 8.
159
Enriched cliques with six vertices
5.
NORr~AL
FORM
Returning to our example (1.5), suppose we omit the X in each row and arrange the entries in each row in decreasing order.
The result is (5.1)
3 3 2 1 0 3 2 2 1 3 2 2 1
322 1 3 2 2 1
33210 This tells us at a glance that vertices 1;6 of the corresponding E-clique have the same type, and that vertices 2,3,4,5 also have the same type, but the types are different.
The same process, applied to (1.8), yields 33210
(5.2)
33210 322 322 1 3 2 2 1
3 2 2 1
Comparison of (5.1), (5.2) tells us that (1.5), (1.8) have the same type, with vertices differently arranged.
We prefer the arrangement (5.2), and call this
norma 1. Since (5.1) or (5.2) is very bulky, we prefer to denote each vertex type by a single letter, chosen from as many as are needed of a,b,c,d,e,f.
For example,
the type 33210 of a vertex for an E-clique of size 9 is denoted by b. 3 3 3 2 2
However, the type
R.H. Bruck
160
of a vertex for an E-clique of size 13 is also denoted by b; the same letters are used with different meanings for different sizes. two tables at the end of the paper. of the letters as vertex types.
The exact details are given in
One table gives, for each size, the meaning
The other lists the type of each E-clique dis-
played in the catalogue of spread-sets.
This type appears as a six-letter "word",
for example, abcddd.
(5.3)
This word, like all the others, has its letters in alphabetical order. An E-clique of type (5.3) is said to be normal, or in normal form, provided that, for 1 < i < 6, the type of the ith vertex is given by the ;th letter of the word.
That is, vertices 1,2,3,4,5,6 have types a,b,c,d,d,d respectively.
Simi-
larly in general. It is often more convenient to write a type such as (5.3) in a form such as
abcd
3
- but I wanted to avoid exponents in the table.
2 4 Both (1.5) and (1.8) have type b d , but (1.8) is in normal form, whereas (1.5) is not.
2 4
Consider the orbits of size 10, type c d three of these, numbered 26, 27, 28. is in normal form.
According to the table there are
In each case the E-clique in the catalogue
However, this does not specify the E-clique uniquely; it can
have 2
4
/ g
48 / g
different forms, where g is the order of the automorphism group of the E-clique. (For orbits 26,27,28, g
= 2,
2 and 8, respectively; the automorphism group coin-
cides with the stability group in each case.)
Thus given, a spread-set of size
2 4
10, type cd, we cannot identify its orbit immediately by reducing its E-clique to an equivalent normal form.
Nevertheless, the concept of normal form simplifies
the problem of equivalence a great deal. One final remark.
In constructing the catalogue I used a variety of methods
which are probably not worth discussing.
Common to all of them was the necessity
of handling some orbits a dozen times or more before arriving at a complete set.
Enriched cliques with six vertices
161
I did the work by hand; I believe that a machine program which left much less for me to do would have been very expensive.
Now that the catalogue exists, however,
it should prove useful as part of a machine program in further exploration.
162
R.H. Brock
CATALOGUE OF SPREAD-SETS ABBREVIATIONS: If = ordinal number. OL = orbit length. D = degree. AD = anti-degree. For definitions. see section I . SUGGESTION: In most cases, a well-designed sketch of the E-clique will reveal a great deal. SIZE 15--1 ORBIT
SIZE 14--1 ORBIT
X33333 If 01 3X3333 OL 001 33X333 15 333X33 D 6 3333X3 33333X AD --
X23333 If 02 2X3333 OL 015 33X233 8 6 332X33 D 6 5 3333X2 33332X AD --
SIZE 13--2 ORBITS X13333 # 03 lX3333 OL 045 33X232 4 8 1 332X23 D 6 5 4 3332X2 33232X AD 12
X23332 # 04 2X2333 OL 060 32X233 4 6 3 332X23 D 6 5 4 3332X2 2 23332X AD 0
SIZE 12-- 5 ORBITS X03333 If OX3333 OL 33X222 332X22 D 3322X2 33222XAD
05 015 5 2
12 3
X3222.3 # 08 3X3222 OL 060 23X322 2 3 6 1 223X32. D 6 5 4 3 2223X3 3 32223X AD 0
X 13323 # 06 IX3332 OL 180 33X 123 2 6 4 331X32 D 6 5 4 2323X2 32322X AD 2112
X 12333 # 07 lX3233 OL 180 23X322 2 5 4 1 323X22 D 6 5 4 3 3322X2 2 1 33222X AD 1 0
X33222 II 09 3X3222 OL 020 33X222 3 222X33 D 5 9 3 2223X3 3 22233X AD 0
SIZE 11--9 ORBITS X03332 II 10 OX3323 ot 180 33X122 6 5 331X22 D 5 4 3222X2 23222X AD 312211
X31313 # 11 3X31310t 090 13X232 1 4 6 312X23 D 6 5 4 1332X2 31232X AD 24
X11333 If 12 IX3232 OL 360 13X322 1 4 4 323XI2 D 6 5 4 3 2 3321X2 32222X AD 14
Enriched cliques with six vertices
SIZE ll--CONTINUED X32123 # 13 3X3212 OL 060 23X321 4 123X32 D 6 47 2123X3 1 32123X AD 3 13
X23132 # 14 lX2313 OL 060 3lX231 6 3 2 13lX23 D 5 4 3 313lX2 2313lX AD 31 13
X33122 # 15 3X1322 OL 360 31X232 1 3 4 3 13lX23 D 6 5 4 3 223lX2 2ZZ3lX AD Zll Z 01
X133ZZ # 16 1X33ZZ OL 045 33Xl ZZ Z 8 1 331X2Z D 6 4 2 2ZZlX3 2 ZZZ23X AD 1 0Z
X133Z2 # 17 1XZZ33 OL 180 3lX2Z2 4 Z 4 3ZlXZZ D 5 4 3 21 23ZlX2 2 2 Z3ZZlX AD 1 0
X3Z2Z2 # 18 3X2ZZ2 OL 030 2lX3Z2 1 6 4 2Z3XZZ D 6 4 3 ZZ2lX3 4 2ZZZ3X AD 0
X03133 # 19 OX332Z OL 360 33X211 13lX2Z D 53 4 6 3 1 3Z1lXZ 2 3Z1ZlX AD 3 Z3
X033Z2 # 20 OX33ZZ OL 045 33XOZZ Z 8 33OXZ2 D 5 4 ZZZlX2 1 ZZZZlX AD 4 Z4
X0332Z # Zl OX3Z3Z OL 360 33Xll Z 3Z1XZZ D 53 4 4 3 3 Z31lX2 Z 1 2 2Z22lX AD 3 2 1
X023Z3 # ZZ OX3232 OL 180 23X221 2 6 Z 32lX1Z D 5 4 3 2321XZ 1 2 Z 3Z1ZlX AD 4 Z 1
X33112 # 23 3X3Z11 OL 060 33X1 Zl lZ1X33 D 53 4 6 3 1 llZ3X3 Z 3 Z1133X AD 3 Z
X13312 # Z4 1X133Z OL 072 31X132 5 5 331X12 D 5 3 1331X2 5 ZZZ2lX AD 2
X3311Z # 25 3XllZ3 OL 360 31X3Z1 1 1 5 3 113X3Z D 6 5 4 3 1223XZ 1 3 1 23122X AD 3 2 1
X33112 # Z6 3X1321 OL 360 31X22Z 1 1 4 3 13lX2Z D 6 5 4 3 21 122lX3 3 2 21223X AD 2 1
X1133Z # Z7 lX3123 OL 360 13XZ2Z 3 3 3 1 31lXZZ D 5 4 3 Z 322lX1 1 13 Z3Z21X AD 3 Z 1
X23113 # 28 lX1331 OL 090 31X222 4 6 13lX22 D 5 3 132lX2 4 1 3122lX AD 2 0
X33112 # 29 3X1222 OL 360 31X222 2 4 2 2 12lX32 D 5 4 3 2 1223X2 2 1 2Z22lX AD 221 0
X32221 # 30 3X1222 OL 120 21X322 1 3 6 223Xl2 D 6 4 3 2221X3 1 1Z223X AD 3 13 01
SIZE 10--15 ORBITS
163
164
R.II. Bruck
SIZE lQ--CONTINUED X31222 /I 31 3X2122 OL 090 12X322 213X22 D 514 4 34 11 2222X2 4 1 2Z22lX AD 1 0
211 2 0 2
X222Z2 /I 33 lX2222 OL 006 22X222 10 222X22 D 3 222lX2 5 22222X AD 0
X00333 /I 34 OX3222 OL 060 03XZ22 3ZlXll D 56 3Z21Xl 6 32211X AD 3
X0313Z /I 35 OX331 Z OL 360 33Xl11 2 3 4 131X2Z D 5 4 3 311lXZ 4 2 Z21 ZlX AD 3 Z
X30312 /I 36 3X30Z1 OL 360 03XZZ2 1 5 3 302XZ2 D 5 4 3 1222X2 21222X AD 4 13 2 2 3
X02133 /I 37 OX3321 OL 360 Z3X211 1 5 3 13lX12 D 5 4 3 nl1X2 1 Z 3 3112lX AD 4 3 2
X0322Z II 38 OX232Z OL 360 32XllZ 1 Z 5 1 231XZ1 D 5 4 3 2 ZZI2XZ ZZ212X AD 4 1312113
X02331 /I 39 OX3Z22 OL 360 23X1l2 321X12 D 5 2 4 2 34 21 3211X2 12Z22X AD 33 2 2 11
X03132 II 40 OXZ322 OL no 3lX211 4 3 13lX12 D 514 3 21 3Z11X2 ZZ122X AD 4 1312 3 11
X33111 II 41 3X3111 OL 010 33Xl11 lllX33 D4 9 1113X3 6 11133X AD 1
X33111 /I 4Z 3Xll13 OL 180 31XZ21 112X32 D 614 3 34 21 1123X2 2 4 13122X AD 3 Z
Xl3311 II 43 1Xl133 OL 180 3lXZ21 Z Z 4 1 312X12 D 5 4 3 Z 132IX2 3 Z1 1312lX AD 3 2 1
X33111 II 44 3XOZZ2 OL 120 3OX222 6 3 122X22 D 5 2 1222X2 3 1222lX AD 3 3 1
X13311 II 45 1Xl Z3Z OL 360 31X122 3Z1X12 D 5 2 4 2 3 2 2 3 13Z1XZ lZZ22X AD 3 12411
X03ZZ2 /I 46 OX23Z2 OL 090 3lXOZ2 23OX2Z 22Z2Xl 2Z221X
X0322Z II 47 OX23Z2 OL 360 32XllZ 5 Z31XZI D 514 2 3 21 Z212X2 Z2212X AD 4 1312113
X32211 /I 48 3X1122 OL 180 ZlX321 Z13X12 D 614 13 6 21 12Z1X3 2 Z2 lZ123X AD 3 2 1
32 180 1 4 3 2 5 4 3 Z
SIZE 9--21 ORBITS
Enriched cliques with six vertices
SIZE 9--CONTINUED X32112 # 49 3X1221 OL 060 21X321 1 3 2 123Xl2 D 6 4 3 2 3 1221X3 21123X AD 312411
X31122 # 50 3Xl122 OL 090 llX322 113X22 D 45 3 2 2 2 222ZXl 22221X AD 3 2 14
X32112 # 52 3X1221 OL 360 21X222 3 3 2 12ZX22 D 4 3 2 11 122ZX2 2122ZX AD 23 12 01
X12222 # 53 lX2222 OL 015 2ZX122 8 221X22 D 3 01 222ZXl 6 22221X AD 1
X31122 # 51 3X1212 OL 360 l1X322 D 5 14 2 3 3 2 3 123X21 212ZX2 2221ZX AD 31231101 54 060 6 3 3 2 3 113 0 2
SIZE 8--24 ORBITS X00332 # 55 OX3212 OL 360 03X122 D 443 4 321XlI 3121Xl 22211X AD 4 2 3 4 21
X33002 # 56 3X0221 OL 180 30X221 4 02ZX22 D 443 022ZX2 4 2112ZX AD 4 2 3 21
X30311 # 57 3X3011 OL 180 03X122 1 2 5 301X22 D 5 4 3 112ZX2 1 6 1122ZX AD 4 3
X01133 # 58 OX3311 OL 090 13X211 4 13ZXll D 443 3111X2 4 3111ZX AD 4 2 3 21
X33011 # 59 3X0212 OL 360 3OX221 2 2 02ZX22 D 443 2 112ZX2 1212ZX AD 4 2 3 2 2 3
X03131 # 60 OX2312 OL 720 32Xll1 1 2 3 2 131X12 D 5 4 3 2 3111X2 1 4 2 1212ZX AD 4 3 2
X20312 # 61 ZX3021 OL 360 03X122 301X22 D 43 3 421 122ZXl 21221X AD 5 13 3 2 3
X20312 # 62 ZX3021 OL 180 03X212 1 2 3 2 D 5 4 3 2 30ZX21 121 ZX2 1 4 2 2121 ZX AD 4 3 2
X20312 # 63 ZX30210L 180 03X221 1 7 30ZX12 D 5 3 1221X2 4 21I2ZX AD 4 3 2
X03122 # 64 OX2312 OL 720 3ZXlli D 514 13 3 2 3 131X21 211 ZX2 1 3 2 2211ZX AD 4 3 2 11
X03122 # 65 OX1322 OL 360 31X121 D 4 3 322 3 131XI2 2221Xl 2 1 22121X AD 4 3 2 311
X03122 if 66 OX1322 OL 180 31X211 1 61 13ZXll D 5 3 1 2211X2 2 2211ZX AD 4 2 5
165
R.H. Bruck
166
SIZE 8--CONTINUEO X30212 # 67 3XZOZl OL 360 OZX2ZZ ZOZX2Z 0 4 2 3 4 22 12ZOO ZI Z21X AD 513 2 2 3 11
X30Z21
X31121 # 70 3XI112 OL 180 11X31 Z 113X21 051422411 211 ZX2 12Z1 ZX AD 3 2 2 5
X31121 # 11 3Xl1 ZI OL 180 11 X312 113Xl Z D 5134 2211 2211X2 1122ZX AD 3 4 2112
X31112 # 72 3X1l21 OL 180 11X222 2 4 llZX22 04 31Z 11 122ZXl 2 4 1 21Z21X AD 3 Z 0 XOZ222 # 75 OXZ22Z OL 090 ZZXI1Z 4 221X21 D 3 24 221ZXl 22Z11X AD 4 Zz114
#
3X2111 OL
68
no
OZXZ22
ZI 00 2 D 4 2 3 32211 2121X2 112ZZX AD 413 2 2 212
73 360 2 4 2 3 2 2 12 Z Z 3 2 31 X1l222 # 76 lXl Z22 OL 060 llX222 22ZXli 2221Xl 22211X 78 120 3 4 2 3 01
1 3 3 3 2 1
X31112 # 69 3XZlll OL 120 1 ZX311 113X21 D 613 4 2 3 l11ZX3 4 21113X AD 3 23
X02222 # 74 OXZ222 OL 015 2ZX022 22OX22 222ZXO 22220X
Enriched cliques with six vertices
167
SIZE AGAINST LETTER TYPES OF VERTICES
15 14 13 12 11
10 9 8
a
b
33333 33332 33331 33330 33320 33310 33300 33200
33322 33321 33311 33220 33210 3311 0
c
e
d
33222 33221 33211 33111 32210
32222 32221 32220 32111
22222 32211 22220
f
22221 22211
TYPES OF CATALOGUED E-CUQUES SIZE
#
TYPE
15
01
aaaaaa
14
02 aaaaaa
13
03 aabbbb 04 bbbbbb
12
05 06 07 08 09
aacccc bbbbcc bbcccc cccccc cccccc
10 11 12 13 14 15 16 17 18
aaccdd bbcccc bccccd cccccc cccccc ccccdd ccccdd ccdddd dddddd
11
SIZE
#
10
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
abcddd bbbbee bbcdde bbdddd cccccc ccccce ccccdd ccdddd ccdddd ccdddd cdddde dddddd ddddee ddddee eeeeee
9
34 35 36 37 38 39
addeee bbceef bbddff bbeeee bbeeff bdeeef
8
9
TYPE
SIZE
# TYPE 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
bdeeef cccccc cceeee cceeee cddfff ceeeef ddddff ddeeff eeeeee eeeeee eeeeff eeeeff eeffff ffffff ffffff
55 56 57 58 59 60
accddf acceef bbccff bbdddd bcceff bcdddf
SIZE
#
TYPE
8
61 62 63 64 65 66 67 68 69 70
ccccff ccccff ccccff ccddff ccddff ccddff cceeff cdefff dddddd ddddff ddddff ddffff ddffff eeeeee eeffff ffffff ffffff ffffff
71 7Z
73 74 75 76 77
78
R.H. Bruck
168
BIBLIOGRAPHY 1. 2. 3.
M. Hall, Jr., Projective planes, Trans. Amer. Math. Soc. 54 (1943), 229-277. R.H.F. Denniston, Non-existence of a certain projective plane, J. Austral. Math. Soc. 10 (1969), 214-218. F.J. MacWilliams, N.J.A. Sloane and J.G. Thompson, On the existence of a projective plane of order 10, J. Combinatorial Theory Ser. A, Z4 (1973), 66-78. Van Vleck Hall University of Wisconsin Madison, WIS. 53706, U.S.A.
Annals of Discrete Mathematics 18 (1983) 169-176 © North-Holland Publishing Company
169
BLOCKING SETS IN AFFINE PLANES :,{
A.A. Bruen and M.J. de Resmini
SECTION 1 As usual AG(2,q) (PG(2,q»
denotes the affine (projective) plane over GF(q)
the finite field of order q. Let n be a finite projective plane of order n. in
n
is a subset S of the points of
n
Recall that a blocking set
satisfying the following two conditions:
(i) Each line of
n
contains at least one point in S;
(ii) Each line of
n
contains at least one point not in S.
We say that S is irreducible if no proper subset of S is also a blocking set.
This is equivalent to saying that each pOint P of S lies on at least one
tangent to S. The following result is contained in [2) and [6]. THEOREt~
1.1: If S is any blocking set in n then
n+
v'n + 1 .;;;
IS I .;;; n2 - y'i\:
Moreover, if S is irreducible then
IS I .;;; n \I'll +
1.
Let n be a finite affine plane of order n.
A (intersection set) in
We define a blocking set
to be a subset Wof the points of n A such that each line A of n A contains at least one point of W. (Note that, unlike the definition in the projective plane, Wmay contain a n
1i ne. )
As in the case of a projective plane Wis irreducible if each point lies on at least one tangent to W. The following is shown in [5] and [7]. THEOREM 1.2: If Wis any blocking set in an affine plane
n
A
of order n then
170
A.A. Bruen andM.J. de Resmini
lWI;;. n
+ vh+
MoY'eover, if W is irreducible then IWI .;; n vh~
2.
In [101 Jamison obtains a surprising and substantial improvement on the lower bound for classical planes, as follows. THEOREM 1.3: Let
rr
A
= AG(2,q) and let Wbe a blocking set in
rr
. A
Then
II>I! ;;. 2q - 1. We should mention here that a somewhat shorter proof of Theorem 1.3 is given in [1 ). On page 254 of his paper Jamison remarks that despite "strenuous efforts" no combinatorial proof of Theorem 1.3 could be found. In r 4) we obtain new bounds on the size of irreducible blocking sets in the classical projective planes which supplement the lower bound of Theorem 1.1. method of proof uses Jamison's result.
The
The brazen idea then occurred to us that
Jamison's bound might not in fact be valid for arbitrary affine planes and that, moreover, the methods in [4) might yield some information on this. In fact this turns out to be the case! Our main result, then, is as follows. THEORH4 1.4: TheY'e exist finite affine planes lIA of order nand bloeking sets W in
rr
A
with IWI .;; 2n - 2.
SECTION 2 To justify Theorem 1.4 we offer the following two constructions. :ONSTRUCTION 1: This construction is based on the simplest case of Theorem 2.2 in 4), i.e. the case when
= l.
~
The construction can be generalized for all 'e only discuss the case
~
= 1.
~
> 1, but for simplicity here
The setting is this.
S is an irreducible
locking set in a finite projective plane of order n, with lSI = n + k. Since 5 is a blocking set no line of et us assume that some line
~
of
11
rr
can contain more than k points of S.
contains exactly k - 1 points of S.
Next, let P be any point of £ not in 5, that is P is in £ \ S. ines of
11
through P other than
£
Of the n
exactly one of them, say x(P), contains pre-
Blocking sets in affine planes
171
cisely two points of 5 while the others contain just one point of 5. We now have the following result. THEOREM 2.1: (aJ If no two of the lines x(P) meet in a point of 5, then 151 ~ n + (n + 3)/2. (bJ In particular, if rr = PG(2,q), then 151 ~ q + (q + 3)/2. (cJ Suppose that for some two points P ,. Q of £, \ S the lines x(P) and x(Q) intersect in a point of 5. Then there exist an affine plane rrA of order n and a ~
blocking set Win rrA with IWI
2n - 2.
PROOF: We consider the two possibilities. Possibility 1.
The line x(P) never meets x(Q) in a pOint of S.
I£,\ S I = (n + 1) - (k - 1) Thus lSI is 151
~
~
=
n - k + 2, it follows that IS I
2n - k + 3.
By definition lSI
n + (n + 3)/2.
This proves (a).
Possibility 2.
~
k - 1 + 2(n - k + 2~
= n + k. Therefore, k ~ (n
For some two points P ,. Q on
9., \
Then, since +
3)/2, that
S the lines x(P), x(Q)
intersect in a point Z of S. Now, S being irreducible, there exists a tangent line z to S at Z.
Let J
denote the set of those lines of rr other than z which pass through Z and which are tangent to S at Z. The lines ZP IJI
~
= x(P)
and ZQ
= x(Q)
are not tangents to S.
Thus
(n + 1) - 1 - 2 - (k - 1) = n - k - l. Let rrA be the affine plane obtained by removing z from rr.
We construct a
set Wof points in rrA as follows. On each line a of J choose anyone point R = R(a) with R(a) ,. Z. H the set of all such points R(a). Finally, set W= (5 \ {Z})
U
tha t Wintersects all 1i nes in rr A' That is IWI
~
2n - 2.
Thus IHI H.
= IJI
~
Denote by
n - k + 1.
Since S is a blocking set in rr it follows
r~oreover,
IWI
~
(n +
k) -
1 + (n - k - 1).
This proves part (c).
It follows from Jamison's result that Possibility 2 cannot occur if n
= PG(2,q). Part (b) now follows. In fact, equality can occur in part (c) of Theorem 2.1.
This will come out
in the proof of the next result. Let K be a k-arc in a finite projective plane of order n. of k points no three collinear. (k + J)-arc.
to K.
A point P of
rr
Thus K is a set
K is complete if K is not contained in any
is said to be exterior to K if P lies on no tangent
172
A.A. Bruen and M.J. de Resminl
THEOREM 2.2: Let in
71.
Let
R.
(a) If
71
b", a proojective pZane of ol'dero 9 and Zet K be a compZete 6-aroc
bl!; a secant of K. R.
Then
contains exactZy one (zero) exterior point, then theroe exist an
affille pZane of ordero 9, "A' and a bZocking set
in 7IA with I WI
= 16 (15).
(b) AZZ known non-Desaroguesian proojective pZanes " of oroder 9 contain a o-aroc Kand a secant
R.
of K which contains exaatZy one exteroior point.
PROOF: Part (b) follows from Dennistons's paper [9) which contains a very useful and detailed discussion of arcs in planes of order nine. Since K is complete. each point P Moreover, P 1ies on at most -k < n + 1
To see Part (a) we proceed as follows. of
71
lies on at least one secant of K.
secants to K.
It follows that in the plane
tn
dual to " the set of = 15 secants to K yields a blocking set S with IS I = 15. Since each point P of K 71,
=9 +
lies on exactly 5 secants it follows that lSI containing exactly 6 - 1
6 with some line of",
= 5 points. It is easy to show that S is irreducible.
Thus we are in the situation of Theorem 2.1. Elementary ad hoc counting arguments show that the hypotheses on the exterior pOints in part (a) guarantee that Possibility 2 of Theorem 2.1 occurs, and yield the results in Part (a). A specific example for the Hughes plane of order nine will be given in Section 3. CONSTRUCTION 2: A celebrated conjecture of Hanna Neumann [11) is that all finite non-Desarguesian projective planes of order n, say, contain projective subplanes of order two.
Of course, if
71
= PG(2.q)
then
subplane of order 2 when q is a power of 2.
71
only contains a projective
However the conjecture is true for
all the known non-Desarguesian planes of order nine, all of which contain projective subplanes of order two. We have exploited this fact in the case of the Hughes plane" of order nine to obtain a blocking set V in an affine plane "A of order 9 obtained by deleting a certain line of THEOREI~
71.
In fact. we have
2.3: The bZocking set V has 16 points.
It contains a Fano subpZane and
Blocking sets in affine planes
has seven points on a line.
173
Moreover, if we construct the blocking set Wof
Theorem 2.2 using for rr the Hughes plane of order nine, then Wcan be chosen so that at most five points of Ware on any line.
SECTION 3 Let rr be the Hughes plane of order nine.
We use Denniston's notations in
[ a ] and [9] for poi nts and 1i nes in rr. Firstly, we construct the blocking set Wof Theorem 2.2 starting from the "non -Desarguesian" complete 6-arc K (wl in [9]) whose points are: F, G, H, N , 3 03' Y4· Then the 15 secants to this arc are the pOints of the blocking set S in the projective plane n, dual to rr.
Namely they are
s, r, fa' f6' f4' t, 96, 9a , 94 , h3' h6' y, i 3 , k7, d7· Next, we choose for the five points on a line
2
(namely the line F of rr,)
the set s, r, fa' f6' f 4· The two lines Pa and W6 throu9h the points p and w, respectively, on F have two points in common with S and meet at the point 9 in S. 4 Now we delete 94 and take Na as the line at infinity. The tangents to S at 94 other than Na are U7 and 07 and we adjoin the point a4 on 07 and the point e on 3 U . Therefore the blockin9 set Wconsists of the followin9 16 pOints of rr, \ Na: 7 s, r, fa' f6' f4' t, 96' ga' h3' h6' y, i 3 , k7' d7, a4 , e3 · Col linearity relations' amon9 the points of Ware displayed below.
At most five
points of Wlie on any line. s, r, fa' f6' f4 r, t, h3' h6' Y f6' 9a , h6' i 3 , d7 fS' t, d7 , e3
on F;
f4' 9S ' e3 s, h3 ' d 7 r, 9 ' k7 S f6' t, k7
on G;
s, t, 96 , 9S fS' 96, h3,i 3 , k7 s, h6' k7' a 4
on N3 ; on S4;
on Za; on S7;
f6' y, 96' a4 , e3 f4' h3' a4 r, 96 , d7
on Y6; on Q6; on R4 ;
on R5; on T5;
fS' 9S' Y f4' t, i 3
on Y ; a on T7 .
on H; on °3; on T3;
Next we describe the set V of Theorem 2.3 in the HU9hes plane nine. ing:
We start from the Fano subp 1ane
C1
11
of order
in [S] whose incidences are the follow-
174
A.A. Bruen and M.J. de Resmini
n
A D N , p: D L P , x: A L X , a : A P 53' 3 6 3 3 6 D 53 X6 , £6: L S3 N , i 6 : P3 N6 X6 • 6
d3
The blocking set V will contain the seven points of lines in n. -A
=- \
0
Choosing the line
0
a,
which cover 56
as line at infinity, the blocking set V in
consists of the following 16 points:
These points lie on the following lines of -A: A, D, N , N6 on n; 4 D, L, P3' P on p; 5 H, X , N4 on h4 ; 6 A, l, X , X , X6 on X; 3 4 L, N , 53' on £6; 6 \ H, W ' Z6' X , N , V , P 6 4 6 5 5
A, P , 53' W6 3 D, 53' X6 , Z6 N6 , X , P 6 3 l, N4 , W6
on a ; 3 on d ; 3 on i6; on £4;
T , P , X , N , V5 5 3 3 4 on h . 6
on m ; 7
Note that the line h6' containing seven points of V, is tangent to the subp1ane
a
at the point N . 6 CONCLUDING REMARKS Despite the examples in sections 2 and 3, it is still not clear how good the lower bound of Theorem 1. 4 rea lly is.
It may be poss i b1e to buil d up "small"
blocking sets in affine planes by starting from subp1anes by analogy with Construction 2.
We are also examining the possibility of making more use of
Theorem 2.1 to construct smaller blocking sets. ACKNOWLEDGEr~ENT:
Research supported in part by the N. S. E. R. C. of Canada and the
C.N.R. of Italy.
A portion of this work was carried out in conjunction with the
Combinatorial Geometries Seminar of Prof. G. Ta11ini in Rome.
We wish to thank
Prof. G. Tallini and the other participants for stimulating discussions. BIBLIOGRAPHY 1. 2.
A.E. Brouwer and A. Schrijver, The blocking number of an affine space" J. Comb. Th. A, 24 (1978), 251-253. A.A. Bruen, Baer subplanes and blocking sets, BuZl. Amer. Math. Soc., 76 (1970), 342-244.
Blocking sets in affine planes
3. 4. 5. 6. 7. 8. 9. 10. 11.
175
A.A. Bruen, Blocking sets in finite projective planes, SIAM J. Appl. Math., 21 (1971), 380-392. A.A. Bruen and R. Silverman, Arcs and blocking sets, in "Finite Geometries and Designs", London Math. Soc. Lecture Note Series 49, Cambridge U.P., 1981, 52-59. A.A. Bruen and R. Silverman, Arcs and blocking sets II, preprint 1981. A.A. Bruen and J.A. Thas, Blocking sets, Geom. Dediaata, 6 (1977), 193-203. A.A. Bruen and J.A. Thas, Irreducible blocking sets, preprint 1981. R.H.F. Denniston, Subplanes of the Hughes plane of order 9, Proa. Cambridge Phil. Soa., 64 (1968), 589-598. R.H.F. Denniston, On arcs in projective planes of order 9, Manusaripta Math., 4 (1971), 61-89. R.E. Jamison, Covering finite fields with cosets of subspaces, J. Comb. Th. A, 22 (1977), 253-266. H. Neumann, On some finite non-Desarguesian planes, Arah. Math., 6 (1955), 36-40. Department of Mathematics Middlesex College The University of Western Ontario London, Ontario, Canada N6A 5B7
Istituto Matematico "G. Castelnuovo" Universita di Roma 00185 Rome, Italy
This Page Intentionally Left Blank
Annals of Discrete Mathematics IS (1983) 177-lS0 © North-Holland Publishing Company
177
SOME BUILDING GEOMETRIES OF SPHERICAL TYPE AS SETS OF POINTS AND LINES F. Buekenhout
My starting point is the following list of Dynkin diagrams or rather Coxeter diagrams of spherical type. A @--o--o
0-----0---0
n
Bn,C
n
@--o---o
...
(}---{)==O
Dn
~
...
E3
~
~
E4
n vertices, n
~ E5
~
1
n vertices, n ,;;; 2
n vertices, n';; 4
®---L--o--o
E6
@-~
E7~E8~ F3 @==o--o
F4 @-----o==o-----
Each of these diagrams determines a class of buildings (and weak buildings) which has been completely classified to a large extent (see Tits (5)). such a building then
f
If r is
and the circled dot of the diagram determine uniquely a
geometry consisting of points and lines in f.
My purpose is to give a set of
axioms characterizing these geometries. A great deal of inspiration, motivation and ideas expressed here are due to various people, mainly J. Tits [5) and references given there), B. Cooperstein [3) and for his study of geometries of type En based on points and lines, A. Cohen (2) for his characterization of geometries of type F4 and Cn, P. Cameron [1) for his characterization of dual polar spaces and G. Tallini (4) for his characterization of Grassmanians of lines of projective spaces.
The author is especially grateful
to J. Tits and W.M. Kantor for useful discussions on exceptional geometries. LIST OF AXIOMS (0) f
= (P,L)
is an incidence structure consisting of a set of points and a
F. Buekenhout
178
family L of subsets of P called lines each of which contains at least two points. Points which are together on some line are called collinear. (1) if L is a line and p is a point then p is collinear with 0,1 or all points of L.
(2) any pair of points is contained in at most one line. DEFINITION: A subspace of r is a set of points S such that every line with two points in S is included in S. A singular
sl~space
of r is a subspace all of whose points are pairwise collinear.
If p,q are points a path from p to q is a sequence of points p=X ,Xl , ... ,X =q in o n which any two consecutive elements are collinear. A convex subspace S of r is a subspace such that for any pair of points p,q of S all shortest paths from p to q in r are contained in S. (3) Every singular subspace is a projective space. (4) If p,q are points of r at distance 2 then either the convex subspace generated by p and q is a polar space of rank n where n
~
2 is the same integer
for all such pairs (and is called a hyperZine) or is the union of two lines intersecting in a point pq in which case (p,q) is called a special pair. (5) P is connected for linearity. (6) If (p,q) is a special pair then there are lines P' ,Q' on pq such that, , 15, which suggests the second question. Can one easily compute the chromatic index of a STS? Of course, an affirmative answer to the second question might significantly simplify the first. We focus on the second question, developing heuristic methods for computing the chromatic index of STS via greedy methods and hill-climbing.
Testing these
methods on the eighty STS(15) suggests that they provide useful approximations to the chromatic index. 2. EXACT COMPUTATION In the case of graphs, Vizing's theorem [18] guarantees that a graph with maximum degree d has chromatic index d or d+1.
Arjomandi [1] gives a clever
polynomial-time method for constructing a (d+1}-co10uring.
Nonetheless, deciding
whether the chromatic index is d or d+l has recently been shown to be NP-complete [10] .
Similar results are completely lacking for STS.
The complexity of computing
the chromatic index is unknown, and the best current method involves (possibly exponential) backtracking.
An alternate direction is to consider algorithms which
always run in polynomial time, but only approximate the answer.
This is a reason-
able compromise in the context of the original tournament scheduling problems. We investigate two general classes of algorithms for approximating the chromatic index: greedy methods and hill-climbing methods. parison is the optimal co10urings of the eighty STS(15}.
Our standard of comFour have chromatic
index 7, thirteen have 8, and the remaining sixty-three have 9 [3]. 3. ACHROMATIC INDEX A simple greedy colouring technique operates as follows. block is a different colour.
Initially, each
Until there are no two disjoint colour classes,
join together two disjoint colour classes (eliminating a colour).
Although easy
to implement, this method may require many colours; we term the maximum number of colours the achromatic index of the STS.
(This use of "achromatic" is from
203
Greedy colouring of Steiner triple systems
Harary and Hedetniemi [8].)
The worst-case performance of this greedy method is
described in two lemmas. LEMMA 3.1: The achromatic index of an STS(v) is at most cv 3/ 2, for c a fixed constant.
PROOF: Suppose we have a t-block colouring; the i'th colour class contains k(i) blocks.
For each i, 3k(i)(v-l)/2
intersect.
Thus ifl k(i)
~
= t-l, since each pair of the t classes
~ = 2t(t-l)/(3(v-l)).
Since iIl k(i)
= v(v-l)/6,
t(t-l) ~ = v(v-l)2/2, which establishes the lemma. LEMMA 3.2: There exist infinitely many STS whose achromatic index is at least cv 3/2, J+'or C a f·,/-xed constant. PROOF: Take a projective plane of order 2(2t+l) __ that is, with block size 2(2t+l) + 1.
Desarguesian planes of these orders exist [7]; moreover, the plane
properties ensure that every two blocks intersect.
Now replace each block of the
plane by a KTS on those elements; these exist since the block size is 3 (mod 6). The resulting system is coloured by assigning each parallel class of each KTS a unique colour. 4. BLOCK-BY-BLOCK GREEDY METHOD The achromatic index is an indication of how badly one can do. sensible "block-by-block" method proceeds as follows. resent colours. Initially, the blocks are not coloured.
A more
We employ integers to repWe colour the blocks one
at a time, assigning a block the least integer so that the resulting colour class contains disjoint blocks.
It is immediately clear that this improves on the ear-
l i er method. LEMMA 4.1: The block-by-block greedy method uses at most 3v/2 coZours for an STS(v). PROOF: Suppose to the contrary that the number of colours exceeds 3v/2. Thus, at some interim stage, a block {x,y,z} could not be assigned to any of the first 3(v-l)/2J colours.
Then {x,y,z} intersects each of these colour classes; but
204
C.J. Co/bourn and M.J. Co/bourn
each of {x,y.z} appear in only (v-l)l2blocks.
This contradiction establishes the
desired result. At first sight. one might expect that an improvement on this crude argument might give a better upper bound. but this is not so. LEMMA 4.2: The block-by-block gl'eedy method may l'equil'e 3[ (v-l)/2J -2 coloUr's. PROOF: Select a KTS(n) with elements v(l) •...• v(n).
Construct an STS(2n+l) with
elements {x(l) •... ,x(n).y(l) •...• y(n).@} as follows.
For each block {v(i).v(j).
v(k)} of the KTS, we have blocks (x(i).x(j).x(k)}. (x(i).y(j).y(k)} • {y(i).x(j),y(k)} and {y(i),y(j).x(k)}.
In addition. we have b10cks{@.x(i).y(i)}
for each i. To colour the STS, we first order each block of the KTS arbitrarily; for example. the block {x.y.z} may be represented as<x.y.z>.
Let r
=
(n-1)/2 and let
V(l), ... ,Vir) be the parallel classes of the KTS. in which the blocks are For each ordered block
0) points
collinear with both. REMARKS: (a) If in axiom (iii) 0 does not occur, then (iv) is an immediate corollary of the other axioms and
~
= (t+l)a.
In such a case S is called a par-
tial geometry [1]. (b) One proves that the dual structure of a semi partial geometry is again a semi partial geometry iff s=t or S is a partial geometry [3].
230
F. De Clerck and I.A. Thas
1.2.
A (semi) partial geometry S is said to be embedded in a projective space
PG(n,q) if the line set of S is a subset of the line set of the projective space and if the point set of S is the set of all points of PG(n,q) on these lines (then s=q). (n
~
All partial geometries embeddable in a projective space PG(n,q)
2) are known: see [2J for the case a = 1 and [9J for the case a> 1.
We
remark that also all partial geometries embeddable in an affine space are known [15J.
However, up to now only the semi partial geometries embeddable in PG(2,q),
PG(3,q), AG(2,q), AG(3,q) are classified [4J [5).
The main difficulty here is
that by axiom (iv) one cannot use induction for the embedding problem in higher dimensions. 1.3.
A (finite) (O,a)-geometry (a> 1) is a connected incidence structure
S=(P,B,I) with a symmetric incidence relation satisfying (ii) and (iii). REMARK: Let x and x', x # x', be collinear points of S, and denote by t+l (resp. t'+l) the number of lines of B incident with x (resp. x').
Then counting in
different ways the number of ordered pairs (U,U'), xIU, x'IU', xfU', x'{U, with U and U' concurrent lines of B, we obtain t(a - 1) = t' (a - 1).
Hence t=t'.
By
the connectedness of S we now see that every point of S is incident with t+l lines of B. 1.4 EXAMPLES OF (O,a}-GEOMETRIES EMBEDDABLE IN PG(n,q) Clearly a (O,a)-geometry S is said to be embedded in PG(n,q) if the line set of S is a subset of the line set of the projective space and if the point set of S is the set of all points of PG(n,q) on these lines. (a) Let U be a set with m elements, U2 = {TcU I iTi=2}, U3 = {TcU II ITi=3}, I the inclusion relation. Then U2 ,3(m)=(u ,U ,I) is a semi partial geometry with 2 3 s=a=2, t=m-3 and u=4.
For certain values of m, these geometries are embeddable
in a PG(n,q).
For example U2 ,3(5) is isomorphic to the Desargues configuration in PG(3,2}, and the geometry U2,3(7) is embeddable in PG(4,2) (8). (b) In PG(n,q) we take a symplectic polarity Wwith rank 2k, i.e. rad W= = II =PG(n-2k,q).
Let P be the set of points of PG(n,q)\ rad W.
Let B be the set of
lines of PG(n,q) which are not totally isotropic in the polarity Wand let I be the natural incidence. t=q
0-1
- 1.
Then W(n,2k,q)=(P,B,I) is a (O,a)-geometry with S=a=q,
231
The embedding afr O,aJ-geametries in PG( n,q)
If k=l, then the geometry W(n,2,q) is a partial geometry and will be denoted by Hqn . If 2k=n+l, so n is odd, then the symplectic polarity is non-singular.
In
this case the geometry W(n,n+l,q) is a semi partial geometry with ~=qn-l(q_l) and will shortly be denoted by ~) [6]. In all other cases W(n,2k,q) is a "proper" (O,a)-geometry. (c) Take a (possibly singular) quadric 0 in PG(n,2), n
~
3.
Let B be the set of
non-intersecting lines of 0, let P be the set of all points incident in PG(n,q) with the elements of B, and let I be the incidence of PG(n,2).
Then (P,B,I) is a
(O,2)-geometry, unless 0 consists of one or two hyperplanes, or 0 is the hyperbolic quadric 0+(3,2) in PG(3,2), or n ~ 4 and 0 is the cone with vertex PG(n-4,2) which projects a hyperbolic quadric of a threespace skew to PG(n-4,2) (in the last two cases the geometry is not connected). +
If n=2d-l and 0 is the non-singular quadric 0-(2d-l,2), then (P,B,I) is a ±
semi partial geometry, denoted by NO (2d-l,2), with parameters s=a=2, 2d-3 d-2 2d-3 d-l + t+1=2 -£2 , ~=2 -£2 , where £=+1 for the hyperbolic quadric 0 (2d-l,2) and £=-1 for the elliptic quadric 0-(2d-l ,2) [16]. If n=2d and Q is non-singular, then (P,B,I) is a semi partial geometry, denoted by NQ(2d,2), which is isomorphic to W(2d-l,2). In all the other cases the geometry is a "proper" (O,2)-geometry. (d) In PG(3,2 h ) (h ~ 2) we take the hyperbolic quadric. Then the same construction as in (c) gives us a (O,2 h- l )-geometry NQ+(3,2 h) with S=2h, t+l=2 h- l (2 h_l). This geometry is never a semi partial geometry. 1.5. We conjecture that the (O,a)-geometries embedded in PG(n,q), but not in PG(nl,q) with nl < n, are the design of points and lines of PG(n,q), or one of the examples in 1.4.
In the paper, part of this conjecture is proved.
2. (O,a)-GEOMETRIES EMBEDDED IN PG(n,q) 2.1.
Suppose S=(P,B,I) is a (O,a)-geometry, embedded in a projective space
PG(n,q), n
~
3, but not in a PG(nl ,q), with nl < n.
Suppose p is a point incident with a line L of S. L containing at least one other line N of S with pIN. not lying in the
plane~.
Let
~
be a plane through
Suppose q is a point of S
By the connectedness of S there is a chain (zo,zl' ...
232
F. De Clerck and l.A. Thas
... ,z u ), with z0 =p, zu =q, and z.1 collinear (in S) with z.1+ l' i=O, ... ,u-1. that
zo""'z~
are in
rr
and that
z~+l
is not in
rr,
£
E
{O, ... ,u-1}.
Assume
Then the line
z£z£+l is denoted by M. Let S' be the connected component of snPG(3,q) containing L,M and N. S' is a (O,a)-geometry embedded in PG(3,q), but not in a plane.
Then
We denote by e
the number of lines of S' through a point of S'. The points and lines of S' in points on a line and paints.
a
IT
constitute a dual Steiner system with q+1
lines through a point, together with a set of m "isolated"
We note that a point x of S' in
of 8' through x is contained in
IT.
rr
is isolated iff no one of the e lines
Hence S' (and also S) has qa-q+a lines in
rr.
It follows that S' has
~ = (qa-q+a)~(e-a}+qa-q+a+me (l) lines. a If
rr'
then by (1)
is another plane defined by two intersecting lines L' and N' of S', IT'
also has exactly m isolated points.
The following lemma is proved by counting the number of lines of S'. LEMMA 2.2: With respect to S', there are three types of planes in the space PG(3,qpS'. (aJ Planes of type (aJ containing qa-q+a lines and p =(q+l)(q-~l)+m a a
points
of S'. (bJ Planes of type (bJ containing exactly one line of 8' and
Pb=Q+1+ 9(q+l!i s - a )(a-1)+m points of P'. (c) Planes of type
(0)
containing no line of 8' and pc_(qa-q+:l(qs+s-a q ) +m
points of P' . Further e(Pb-pc)=q and the number of points v' of S' equals (q+l)pc'
COROLLARY: If there is at least one plane of type (b) and at least one plane of type (c) then slq.
REMARK 2.3: From 2.1 follows that the solution of the general embedding problem depends entirely on the classification of all (O,a}-geometries which are embedded in PG( 3,q).
233
The embedding of (O,a).geometries in PG(n,q)
3. THE EMBEDDING OF (O,a)-GEOMETRIES WITH m = 0 IN PG(3,g) THEOREM 1: Let S=(P,B,I) be a (O,a)-geometry embedded in a PG(3,q), but not in a p~ane.
If for some
p~ane
of type (a) we have m=O, then the
fo~~owing
cases are
possib~e.
a=q+l and S is the design of points and
(1)
of PG(3,q);
~ines
a=q and S=H 3 . q' a=q=2 and S=NQ-(3,2).
(2) (3)
PROOF: Suppose that all lines of some plane of type (a) are elements of B.
This
is equivalent to a=q+l, and then the dual Steiner system defined by any two 2
intersecting lines of S is a projective plane. lines of S equals (q2+1)(q2+q+l ). (q+l)(q2+q+l ).
So the number of points of S equals
Consequently S is the design of points and lines of PG(3,q).
Suppose now that in each plane which is not in B. type (a).
Hence e=q +q+l and the number of
TI
of type (a) there is at least one line L
For brevity we shall say that L is a secant of the plane
Such a secant evidently contains q- 3+ 1 points of S. a
TI
of
If T is the
number of planes of type (a) through the secant L, then there are (q- 3 +l)(e-Ta) planes of type (b) through L (implying e-T~). After some calculations there follows that e=Ta+l. First suppose e=Ta. 3 S=H (e=q2).
a
Hence (q- 3 +l)(e-Ta)~+l-T. a
e-Ta~l+ a~l - (a-l)~q+l) and so e=Ta or
Then S is a partial geometry and by [9] a=S=q,
q
If e=Ta+l, then e
then there is no pZane of type
(b).
PROOF: Suppose that there is at least one plane of each type.
The total number of
planes of type (a) and (b) is less or equal than (q+l)(q2+1).
If we put qa-q+a=6
this means that g:;2 ( a 6
(8-~)+6+
me)(q+l-
e-l
~
+
8-1
6(a-l))~(q+l)(q
2 +1)
(3)
This implies that (qa-Q+a-8+1)(sq - q+ ~)+me (1- ~)~2+1. a a qa-q+a We remark that ale. Indeed, elq (since e(Pb-pc)=q) and alq (the dual Steiner system in any plane prime power.
TI
of type (a) has q-~l points on any secant in a
TI)
with q a
Since there is at least one plane of type (b) we have
a qa-q+a 0 .
Hence
(qa-q+a-e+l) (Eq _ q+ ~)O and
q(a-2)(a-l)(q-a)(-q-l)-aq 2(a-2)2 and
~.
Consequently the only semi partial geometries, with 0>1, embedded in a PG(n,q), ~3>
q>3, but not in a PG(n',q), n' prop . .3. J (ii) is obtained.
,° ,°
PROOF: Assume the existing nuclei are 0 and U = (0,1,1,1) (then 01 will 4 3 2 l be a point whose polar quadric does not vanish); thus ~ is defined by
=
- Exy - Fxz - Gxt + Syz + Vyt - (S + V)zt
°
2
Ex + Sxz + Vxt = 2 Fx + Sxy - (5 + V)xt = Gx 2 - (5 + V)xz+ Vxy = 0,
(4.2)
° 5,V~0[4).
The polar quadrics of the given nuclei split into Ex + 5z + Vt Gx - (5
+
V)z
+
Vy
° °
(E + F + G)x + (S + V)y - Vz - St respectively.
p
(4.4)
11
=
°,
The planes (4.3) have exactly one common point:
x
- SV(5 + V)
y
- E(5 + V)2 - GS(S + V) - FV(5 + V) G5V - FV 2 - EV(S + V)
z
and the planes
0
Fx + 5y - (5 + V)t (4.3)
°
t = S(FV - E{5 + V) - GS). P is a nucleus [4) and the polar quadric of any point belonging to into (4.5)
11
and a plane through p. Syz + Vyt - (5 + V)zt
are nuclei.
+
V#
splits
Furthermore, all the pOints on the conic C
=x =
°
C is a degenerate conic iff p E
avoid this case, S
'IT
11
°will be assumed [4).
and then prop. 3.1(ii) holds.
To
Therefore, the locus of nuclei
consists of p and the points on C, and it is easy to check statement (b) [4}.
°
Taking p as a point 1 , the equation of F is 3 3 3 3 (4.6) Ax + By + Cz + Dt + Sxyz + Vxyt - {S
+
V)xzt
0,
253
On cubic surfaces over a field of characteristic 3
and
meets F at a line counted thrice.
TI
All lines on
are tri-tangent lines and
TI
the same holds for the generators of the cone r projecting C from p.
The polar
quadric of a point on r \ C has rank 3; the polar quadric of any point neither on r nor on
TI
has rank 4.
Conversely, assume a plane
TI
exists meeting F at a line g counted three
times (and no point exists whose polar quadric vanishes). that
= 0,
L
is defined by
+
Tzt
+
+
Vxt
+
Vyt '" 0 Wzt '" 0
is the plane t
TI
Pt 2
(
- Gxt
+
I
- Kyt
+
I
- Nzt + 2 2 Gx + Ky
(4.7) )J
\
Qt2 Rt 2 +
Txt + Wyt '" 0 2 Nz - Pxt - Qyt - Rzt
Taking coordinates so
+
+
Therefore, the polar quadric of any point on
Txz
+
Vxy
+
Wyz '" O.
splits into
TI
TI
and a plane through
the point p whose coordinates are
( I
(4.8)
i
l
2
x '" P(KN - W ) - Q(- VN - TW)
+
R(VW
KT)
+
Y '" - P(- NV - TW) + Q(GN - T2) - R(- GW - TV) z '" P(VW + KT) - Q(- GW - VT) + R(GK _ V2) t = GKN + TVW - KT2 - NV 2 - GW 2
It is easy to check p is a nucleus; moreover, all points on the conic
(4.9)
22 Gx 2 + Ky + Nz
are nuclei.
+
Txz
+
Vxy
+
Wyz '" t
=0
C is a degenerate conic iff p is on
Assume C is non-degenerate.
TI
[4] (then prop. 3.1 (ii) holds).
Taking the canonical form for C and the point p as
0 , the equation of F becomes [4]: 4 (4.10)
Ax
3333 +
By
+
Cz
+
Dt
+
t(Gx
222 + Ky + Nz ) '"
o.
All generators of the cone r projecting C from p and all lines on
TI
are tri-
tangent 1i nes. Since g and C have two common pOints, F always contains two of its nuclei, and either (1) g meets C at two distinct points, or (2) g is tangent to C. (1) Take suitable coordinates on (4.11 )
Cz 3
+
Dt 3
+
t(Nz 2
+
TI
[4]; then the equation of F is
Vxy) = 0 ,
C, N, V
0 and O are binodes sharing the biplane
1 2 and y = 0, taO do not intersect on F.
TI
#0 .
and the biplanes x '" 0, t
Two cases are to be considered.
=0
M. de Finis and M.J. de Resmini
254
(i) The nucleus p is not on F; F has only the two binodes 0 and O2 , 1 (ii) P belongs to F; then F has a conic node besides the two binodes. (i)
Through the line g
=Fn
there are exactly three planes, beside TI,
TI
intersecting F at two lines other than g and each of them contains a pair of lines on the biplanes (F TI); the canonical form for F is [41: 332 t + t(xy + z )
z +
(4.12 )
All the lines on
TI
=0
.
and all the generators of the cone xy + z2
0 are tri-
tangent lines and no other tri-tangent line exists.
= 0 in (4.11). The tangent cone at the conic node p meets F exactly
(ii) 0
at the lines joining p with the binodes. (beside
Through 9 there is just one plane
and the plane through the singular points) meeting F at g and at the
~
lines Cz
(4.13 )
+
Nt
=
x
=
0, and Cz
Nt = Y = 0 ,
+
each of them belonging to one biplane.
F contains no other line.
(2) The point at which g is tangent to C is a nucleus belonging to F and is a binode.
On
(4.9) 141.
Ky2
+
~
coordinates can be chosen to get the canonical form for the conic
The polar quadric of any point on TI has rank 2, that of a pOint on r:
Txz = 0, but not on C, has rank 3.
rank 4.
The polar quadric of any other point has
Two cases have to be considered.
(iii) F has just a binode at the pOint g n C = 0, the biplanes intersect on F (at g) and the biplane other than
TI
meets F at two distinct lines (beside g).
F contains no other lines and the canonical form for F is 3
(4.14 )
z - t
3
+ tty
2
+
xz)
= O.
(iv) F has also a conic node at the point p
= 04 ,
the tangent cone being r.
The only lines on Fare g and the line joining p with the binode.
The canonical
form for F is [4]: (4.15 )
z3 + t(y2 + xz)
= O.
As a corollary to prop. 3.1(ii) and 4.1, the following holds. PROPOSITION 4.16: If foul' coplanar nuclei exist, on a plane TI, three of them !,c;'ng ,":Uinear (on a line
9.)
and no point exists whose polar quadric van1:shes,
On cubic surfaces over a field of characteristic 3
(i) the polar quadric of any point on
TI
splits into
TI
255
and a plane through a
point p on £; (ii) on
TI
there exists a degenerate conic, containing £, all whose points are
nuclei, and no nucleus exists off (iii)
TI
TI;
meets F at a line counted three times and F is one of the surfaces
considered in prop. 3.1(ii).
5. ON CUBIC SURFACES HAVING THREE DISTINCT NON-COLLINEAR NUCLEI Beside characterizing some cubic surfaces, the results in this section will show the role the rank of polar quadrics of nuclei plays. PROPOSITION 5.1: If F has three distinct non-collinear nuclei whose polar quadrics have exactly rank 3 and the plane
TI
joining them does not intersect F at a line
counted three times, then two other nuclei not on (a) the nuclei not belonging to
TI
TI
exist, and either
are distinct, or
(b) they coincide and there is a plane, distinct from
TI,
meeting
F at a line
counted three times so that prop. 4.1 holds. When (a) is true, F has exactly five nuclei (no four Of them being coplanar) and one of the following holds: (i) F contains none of its nuclei (therefore, it has no singular point); (ii) F contains one of its nuclei and so it has a conic node; (iii) F contains two of its nuclei and has two conic nodes; (iv) F contains three of its nuclei; thus it has three conic nodes; (v) F contains four of its nuclei and has four conic nodes. In all these cases the numbers and configurations of lines on the surface are as in the classical case (projective 3-space over the complex number field).
PROOF: Assume the given nuclei are 0 , 0 and O , Thus, 4 3 2 (
- Exy - Fxz - Gxt + Syz + Tzt + Vyt
I I,
Ei + Sxz + Vxt + Wzt Fx 2 + Sxy + Txt + Wyt 2 Gx + Txz + Vxy + Wyz
(5.2) {
a
L
is defined by
=0
0 0 0,
W# 0 [41.
Solving the simultaneous equations (5.2), the following one is obtained:
256
M. de Finis and M.J. de Resmini
2 2 2 t 2 (GW - TVW) + tx(STV - GSW) + x (GS - EST - FSV + EFW)
(5.3)
Therefore, two nuclei not belonging to (a)
The two nuclei not on
TI
TI
=
°
exist, and two cases are to be considered.
are distinct.
Take one of them as 01' then the
other one is (W,T,V,S), and by a suitable choice of the unit point, the equation of F becomes (5.4)
3
3
Ax + By + Cz
3
3
+ Dt + xyz + xyt + xzt + yzt
=
°.
(b) The two roots of equation (5.3) coincide; then there exists a plane (# TI) meeting F at a line counted three times and prop. 4.1 holds.
Thus, assume (a) holds and the equation of F is (5.4).
All possible cases
for F will now be considered (for more details, see [4]). (i) When A, B, C, D # 0, A + B + C + D + 1 # 0, none of the nuclei belongs to F; there are exactly five non-degenerate cones all of whose generators are tritangent to F. (ii) F contains exactly one of its nuclei, thus is has a conic node; assume it is 04; then the eqn. to F is (5.4) where D = 0, and A, B, C # 0, A+B+C+l #0.
there are four cones all whose generators are tri-tangent to F. (iii) Two of the five nuclei belong to F; therefore, there are two conic nodes and three cones all whose generators are tri-tangent to F. (iv) F contains three of its five nuclei; thus it has three conic nodes and there are exactly two cones all whose generators are tri-tangent to F. (v) All nuclei but one belong to F; therefore F has four conic nodes and there is just one cone all whose generators are tri-tangent to F. nuclei on F are the points 0i (i (5.5)
xyz + xyt + xzt + yzt
Assume the
= 1,2,3,4); then the equation of F is =
°.
In all cases (i) - (v) the numbers and configurations of the lines belonging to F are as in the classical case [1 I, [3 I (see also (4)). The canonical forms for the surfaces obtained from (5.4) in cases (i) - (v) are different from the classical ones; the latter will be discussed in sect. 7. REMARK: When (v) holds, in the dual space the planes tangent to F form - as in the classical case - Steiner's Roman surface. The next result shows the assumption on rank in prop. 5.1 cannot be dropped.
257
On cubic surfaces over a field of characteristic 3
PROPOSITION 5.6: If F has
th~ee
non-collinear nuclei, their plane does not inter-
sect F at a line counted three times and the polar othe~
given nuclei has rank 2 (the nuclei.
If the polar
q~ics
qua~ic
of exactly one of the
two having rank 3), then F has just three
of two nuclei have rank 2, then there exists a
conic all of whose points are nuclei and a nucleus not belonging to the plane of the conic, and prop. 4.1 holdS.
Finally, if the polar
nuclei have rank 2. then there exists a point whose
qua~ics
of all three
pola~ qua~ic
vanishes and
prop. 2.5 holdS. Under the assumption just one of the polar nuclei has
~ank
qua~ics
of the three given
2, one of the following occurs:
(i) F has two conic nodes and one binode. and either (a) the line
s at whioh the biplanes intersect belongs to F. or
(bJ s does not belong to F; (ii) F has two conic nodes and there are two pencils of tri-tangent lines through the same point on two distinct planes; (iii) F has a conic node and a binode and there is exactly one cone all whose generators are tri-tangent; either (aJ or (b) in (i) holds; (iv) F has just one oonic node and both a cone and two pencils of tri-tangent lines (as in (ii)) exist; (v) F has just a binode and either (a) or (b) in (i) holds; oones all whose
are two
are tri-tangent to F;
generato~s
(vi) F has no
the~e
singula~
point; all the generators of two distinot oanes and
all the lines of two pencils
(as
in (ii)) are tri-tangent.
PROOF: Take coordinates as in the proof of prop. 5.1; r is defined by (5.2). where Wf O.
Under the further assumption the polar quadric of the point 04 has
rank 2 and splits into the planes y
= 0 and z = 0, r is defined by
Exy - Fxz + Syz = 0
f - Ei
(5.7)
) )
l
+ Sxz + Wzt 2 Fx + Sxy + Wyt
Wyz
0
If either F
1:1
0
°
Wf O.
= 0, or E = 0, then there exist a conic all of whose points are
nuclei and a nucleus not on its plane, and prop. 4.1 holds. E
If both F
0 and
0, then a pOint whose polar quadric vanishes exists and prop. 2.5 holds [4].
M. de Finis and M.J. de Resmini
258
Thus, E, FrO will be assumed. Let s be the line y
Take as 01 the point (W,O,O,-S); then S =
(W,O,O,-S) has rank 3. in eqn.s (5.7).
= z = 0; the polar quadric of any pOint on s but 04 and
The polar quadric of any point on
nuclei and not on the line Ey rank 4.
°
not on a line joining two
= x = (which contains just one nucleus) has
Fz
+
TI
°and E,F,W F °
°3, 04;
The only solutions to equations (5.7) are the given nuclei 02'
therefore. all surfaces for which E is defined by (5.7) (with S
= 0) have exactly
three nuclei, the equation of such an F being Ax
(5.8)
3
+
By
3
+
Cz
3
+
Dt
3
+
2
2
Ex Y + Fx z
+
Wyzt =
°.
(No plane meeting F at a line counted three times exists.) (i) All three nuclei belong to F; then F has two conic nodes (02 and 03) and a binode (04)' (b) s
~
The biplanes intersect at the line s. and either (a) s
E
F, or
F.
(a) F contains exactly five distinct lines [4] (as in the classical case), no proper tri-tangent exists. and the canonical form for F is 2
x (y
(5.9)
+
z)
+
yzt =
°.
(b) Since s does not belong to F. A F
° in (5.8).
Each biplane meets F in
the line joining the binode to the conic node belonging to it (counted twice) and at another line.
The configuration of lines on F is as in the classical case \ 4)
and the canonical form for F is 2
x (x
(5.10)
+
y
+
z)
+
yzt
0.
(ii) F has two conic nodes.
The number and configuration of the lines on
the surface are as in prop. 5.1(iii), but now the tri-tangents to F are exactly all lines through 04 belonging either to the plane y \4].
= 0. or to the plane z =
°
The canonical form for F is 2 x (x
(5.11)
+
y + z) + t(yz
+
2 t ) =
°.
(iii) F contains the nucleus whose polar quadric has rank 2 and one of the other nuclei.
There is exactly one cone all of whose generators are tri-tangent
to F and no other tri-tangent line exists. (a) s
E
F (then A = 0).
One biplane meets F at s counted twice and at
another line, the other biplane meets F at s and at two distinct lines.
Besides
the biplanes. there are three planes meeting F at lines (one of them through s)
259
On cubic surfaces over a field of characteristic 3
[4].
With a suitable choice of the unit point, the equation of F becomes 2
(5.12)
2
2
y(y - z ) + z(Fx + Wyt)
=0
F, W-F 0 .
,
One biplane meets F at the line 0 0 g counted twice and at 34 the line r: Ax + Fz = Y = 0, the other biplane meets F at three distinct lines. (b) s
~
F.
Through g there are three other planes meeting F at lines; also through r there are three other planes intersecting F at lines [4]. 3
(5.13)
3
2
2
x + y + x y + z (x + yt)
The equation for F is
= 0 .
(iv) F contains one of its nuclei having a rank 3 polar quadric; thus, F has just a conic node.
The situation is as in prop. 5.1(i), but now the tri-tangent
lines are all generators of a cone (the polar quadric of O2 ) and all the lines through 0 belonging either to the plane y = 0, or to the plane z = 0 [4]. The 4
canonical form for F can be obtained from eqn. (5.8) (and it is not the classical one) . (v) F contains its nucleus whose polar quadric has rank 2. a binode. (a) s
Then F has just
The tri-tangents to F are exactly the generators of two cones. E
F.
Each biplane meets F at s and at two distinct lines and there is
exactly one other line of F (not through 0 ) incident with s [4].
F contains four
4
other lines and the (classical) canonical form (5.14)
(y + z)(y + z + x)(y + z - x) + Wyzt (b) s
~
F.
is obtained
[4]:
W-F 0
= 0 ,
Then each biplane meets F at three distinct lines [4].
are as in the classical case.) (5.15)
[1]
(Lines
The canonical form for F is
(- x + y + z)((- x + y + z) (vi) F contains none of its nuclei.
2
2 + x ) + Wyzt
=0
All generators of two cones and all the
lines through the same point (0 4 ) on two distinct planes are tri-tangent to F and no other tri-tangent line exists. F has 27 lines. For instance, all the required conditions are satisfied by [4]: (5.16)
3 3 2 3 y - z + x (y - z) + t - yzt = 0
6. ON CUBIC SURFACES WITH EITHER TWO OR ONE NUCLEI
PROPOSITION 6.1: If a cubic surface F has two (distinct) nuclei and no point on
260
M. de Finis and M.J. de Resmini
the Zine joining them is a nucleus, then one of the following holds: the polar> quadJ"ics of both nuclei have rank 3; then there are exactly
(-i)
three other nuclei and either they are distinct and prop. 5.1 holds,
01'
they
coincide at a nucleus whose polar quadric has rank 2 and prop. 5.6 holds; ~he
(ii)
poZar> quadrics of the two nuclei have rank 3 and 2, respectively;
then either 1. they-e is a third nucleus whose polar quadric has rank 3, and prop. 5.6
2. no other nucleus exists;
01'
3. there exists a conic all whose points are nuclei and prop. 4.1 holds. When (ii)2. holds, one of the following occurs:
F contains a binode and a conic node; no tri-tangent line exists not
(aJ
passing through the singular points; the biplanes meet at the line s, which may belong
F
:;0
01'
nor;.
(b) F contains just a binode (and either
S E
F,
01' S
i F) and the tri-
tangent,,: are exactly ,;he gener-ators of a cone. (c) F conr;ains Just a conic node and its tri-tangents ar>e exactly the lines of
~-:,)o
pencils 1.Jith trze same centre on two distinct planes.
F has no
(1)
sil~gular
point and the locus of tri-tangents consists of the
"el1eratcrs of a cone and the lines through a point on two distinct planes. (;:ii) The polar> quadrics of both nuclei have rank 2; then either 1. r;he pair's of planes into which they split are distinct (then a line all 1.Jhose points are nuclei exists and prop. ;).1 ho Ids); or 2. the pairs of planes share a plane (then a conic all whose points are exis~s
nucZei
and prop. 4.1 holds).
PROOF: (i) Coordinates can be chosen so that 03 and 04 are the existing nuclei, their polar quadrics both have rank 3, and no point on the line a nucleus. (
(6.2)
I
) ) I
l
joining them is
Then. with a suitable choice of points 01 and 02 [4]. E is defined by
2 - Exy + Hy + Syz + Tzt + Vyt
0
E/ - Hxy + Sxz + Vxt + Wzt
0
Sxy + Txt + Wyt
° ° The polar quadric of any point on I
t
T, S, V. WF 0 •
Txz + Vxy + Wyz
t but q
(O.O,V.-S) has rank 3.
The
261
On cubic surfaces over a field of characteristic 3
polar quadric of q splits into distinct planes not through
~.
Solving the simultaneous equations (6.2) the following one is obtained: 23 ES x
(6.3)
+
(ESW
+
22 HTS - VS )x t
23 VW t = 0 ;
+
eqn. (6.3) has either three distinct or three coincident roots according to whether (6.4)
ESW
+
HTS - VS 2 I 0 , or
(6.5)
RSW
+
HTS - VS 2
=
0
When (6.4) holds, F has five distinct nuclei (the nuclei given by eqn. (6.3) are non-collinear and belong to the plane Sz - Vt = 0), and prop. 5.l(a) holds. When (6.5) holds, the third nucleus is the point V ~E2W2(EW + HT),
x
= -
z
= EV 2 , t = E(EW
(6.6)
+
Y=
-
ETV
~ HT{EW
= EW +V HT
HT) , where S
+ HT) ,
I 0 ;
its polar quadric has rank 2 and prop. 5.6 holds. (ii) Take as 04 the nucleus having a rank 2 polar quadric, let it be yz = 0, and as 0 the nucleus having a rank 3 polar quadric; then E is defined by 3
(6.7)
( - Exy - Fxz + Hy 2 + Syz = 0 J Ex 2 - Hxy - Jyz + Sxz + Wzt Fx 2 + Jy2 + Sxy + Wyt = °
l
Wyz
=
=
0
0,
F, W I
°,
and no nucleus on the line 03°4 exists. When E I 0, the equations (6.7) have, beside 03 and 04' another solution, namely the point (6.8)
x
= - EHW, Y = - E2W, z = 0, t = FH 2
+
JE
2
+
EHS ,
whose polar quadric has rank 3, and prop. 5.6 holds (see [4]). Now, assume E = 0; then the simultaneous equations (6.8) have just the pOints 03 and 04 as solutions. The choice of the reference simplex can be completed in order to get a canonical form for the polar quadric of 03 [4]. The polar quadric of any point but 04 on the lines xz = y = 0 has rank 3. tion of F becomes [4]:
(6.9)
Ax
3
+
By
3
+
Gz
3
+
Dt
3
and one of the following occurs:
+
2 Fx z
+
Hxy
2
+
Wyzt = 0 ,
The equa-
F,H,WIO,
262
M. de Finis and M.J. de Resmini
(a) both nuclei belong to F; (b) the nucleus 04 belongs to F', (c) the nucleus 0 belongs to F', 3
(a) F has a conic node and a binode; the line s: y = z
°mayor not
belong to F. (a.l) s E F.
One biplane meets F at the line
t
= 0 04 (counted twice) and 3
at S; the other at s (counted twice) and at By + Hx = z = 0, and no other plane through s exists meeting F at lines. lines: FBx - HWt = Hx + By =
°and
The tangent cone at 0
3
intersects F at two
F contains no other line and by a suitable
t.
choice of the unit point, the equation of F is 2
2
=
y (y + x) + z(x + yt)
(6.10)
(a.2) s
F.
~
°
One biplane meets F at t (counted twice) and at the line
Ax + Fz = Y = 0; the other at three distinct lines.
The equation of F becomes
[ 4\: (6.11)
x(x
2
2 - Y + Fxz) + Wyzt =
°.
(b) F has just a binode and the tri-tangents to F are exactly the generators of a cone (the polar quadric of 0 ), 3
(b.1) s E F. One biplane meets F at s and at the two distinct lines 2 2 Cz + Fx = Y = 0, while the other meets F at s counted twice and at By + Hx = z = O. These three lines belong to two planes, namely Hv'Tx+B
VTy~H
each of them containing another line on F. 2
Y (x + y) + z(x
(6.12)
(b.2) s
~
F.
2
~z=O,
The canonical form for F is
2
- z ) + Wyzt =
°.
Each biplane meets F at three distinct lines and F contains 9
other 1i nes [4\. (c) F has just a conic node, 03'
The tri-tangents are exactly the lines of
two pencils with the same centre 04 and belonging to the distinct planes y = z = O.
°and
The lines on F are as in prop. 5.1(ii). (d) F does not contain any of its nuclei.
The locus of tri-tangents consists
of the generators of a cone and the lines of two pencils as in (c). lines on F.
An example of such a surface is given by [41:
There are 27
263
On cubic surfaces over a field of characteristic 3
Cx
(6.13)
3
+
(C
+
W)y
3
+
Cz
3
+
(C
+
3
2
W)t - Wx z
+
Wxy
2
+
Wyzt
=
°.
(iii) Take coordinates so that 04 and 03 are the nuclei and xy = 0, Fx +
2
Jy =
°their polar quadrics, respectively.
Writing down
L
(where F,J,V
2 ~
+
0)
it is straightforward that a line all of whose pOints are nuclei exists, and prop. 3.1 holds
[4].
When the polar quadrics of the nuclei 03 and 04 share a plane, taking coordinates so that these quadrics are yt
=
°and yz
=
0, resp .• equations (1.2)
become - Exy
(6.14)
+
Hy 2
= 0, Ex 2 - Hxy
+
Wzt
= 0, Wyt = 0, Wyz = 0,
where at least one between E and H is not zero.
Then - Ex
2
+
Wzt
=Y
W~ 0,
°
is a non-degenerate conic all of whose points are nuclei and prop. 4.1 holds [4].
PROPOSITION 6.15: If a point p exists whose polar quadric splits into two planes, both of them through p, then p is a nucleus and one of the following holds: (i) there are two other nuclei and both their polar quadrics have rank 3
(then prop. 5.6 holds); (ii) there exists but one other nucleus and its polar quadric has rank 3
(then prop. 6.1(ii}2 holds); (iii) there is a line all whose points are nuclei and prop. 3.1 holds.
PROOF: Take 04 as p and yz = p is a nucleus. y
0, or to z
°as its polar quadric; then it is easy to check that
If another nucleus exists, then it must belong either to
= 0, and one of the following holds [4]:
(i) there are two nuclei, distinct from p, one on y = 0, the other on z
0;
(ii) one of the nuclei in (i) coincides with p; (iii) both nuclei in (i) coincide with p. The statement follows. PROPOSITION 6.16: If a point p exists such that its polar quadric has rank 1, being the plane w counted twice, then p is a nucleus, and either
1.
w intersects
F at a line counted three times, in which case w contains a
degenerate conic all of whose points are nuclei and prop. 4.1 holds; or
2.
w does not meet
F at a line counted three times; then one of the fol-
lowing occurs: (i) p is the unique nucleus; (ii) w contains a line t through pall
264
M. de Finis and M.J. de Resmini
b.'huse points Clr8 nuc:2i and no nucleus off R. exists, and prop. :3·1 (i) holds.
(:,) F has a unique singular point, namely the uninode p, the uniplane being (,; (a'ld eitht.:r w meet.3 F at three distinct lines, or
w
intersects it at two
coincident (mld one c','stinct) Zines); or' Ii,}
F has no s:ngular' point and its tn-tangents are exactly the lines on
PROOF: Taking p as 04 and
as x
w
0,
1F/
(6.17)
i\
Gx2
Thus P
J =
+ Jy
2
is defined by
L
- Exy - Fxz - Gxt + Hy2 + lz2 2 Ex - Hxy - Jyz + Mz2 + Sxz - lxz - Myz + Sxy
w
+
° °
Syz =
°
o,
G 'f 0 .
04 is a nucleus.
If w meets F at a line counted three times, then 2 2 N = 0 in eqn.s (6.17) and the degenerate conic Hy + Lz + Syz = x = 0 is the
locus of nuclei and prop. 4.1 holds. iherefore, assume J and M are not simultaneously equal to zero.
Then,
either p is the unique solution to equations (6.17), or HM2
(6.18)
+
lJ 2
SMJ = 0 ,
+
and the line Jy - Mz = x = 0 belongs to all quadrics in I. 2 (i) HM2 + LJ + SMJ 'f O. The polar quadric of any point on w is a cone with vertex at p and w contains a cubic curve the polar quadrics of all whose points have rank at most 2, and this curve has a cusp at p (the tangent being Jy - Mz
x = OJ.
Taking the following canonical form for this curve [4]: 2 3
2
H Y + GMz t
(6.19)
= x =0
,
H, G, M
and as yz = 0 the polar quadric of 0 ,
I
3
(6.20)
- Gxt + Hy
2
= 0,
- Hxy + Mz
2
'f 0 ,
is defined by
= 0, - Myz = 0, Gx 2 = 0 •
All points in P have a rank 4 polar quadric but the points on the quartic surface (6.21 )
2 2 x (Hxy - Mz ) = 0 ;
the plane w contains the curve (6.19) all whose points but the cusp p have a rank
265
On cubic surfaces over a field of characteristic 3
2 polar quadric and p is the unique point whose polar quadric has rank 1. The equation of F is Ax 3 + By3 + Cz 3 + Dt 3 + Gx 2t + Hxy2 + MYZ2
(6.22)
0,
G,H,M
-f 0 .
All lines through p on ware tri-tangent to F and no other tri-tangent exists. (a) p E F; P lines.
= 04 is a uninode, the uniplane is wand meets F at three
Since M -f 0, either these lines are all distinct and the following equa-
tion for F is obtained [4): By3 + Hxy 2 + Myz 2 + Gx 2t = 0 ,
(6.23)
B, H, M, G -f 0 ,
or two of them coincide and the equation of F is 2 2 2 3 Cz + Hxy + Myz + Gx t
(6.24)
o,
H, M, G -f 0 •
(b) P It F. Since all lines through p on ware tri-tangent to F, coordinates can be chosen so that the equation for F becomes [4): (6.25) All tri-tangents to F are exactly all tri-tangents to the cubic curve F n w. surface contains 27 lines [4). (ii) HM2 + LJ 2 + SMJ = O.
Then all points on the line Jy - Mz = x
The
0 are
nuclei and it is easy to check that prop. 3.1(i) holds [4). PROPOSITION 6.26: If a point exists whose polar quadric has rank 4 and if F has a nucleus whose polar quadric has rank 3 (and not less), then one of the following is true: (i) F has exactly five nuclei (whose polar quadrics all have rank 3) and prop. 5.1{a) holds; (ii) F has another nucleus whose polar quadric has rank 2 and prop. 6•. 1(ii) holds; (iii) there are two other nuclei with rank 2 polar quadrics; then a nondegenerate conic all whose points are nuclei exists and prop. 4.1 holds.
PROOF: No pOint exists whose polar quadric vanishes.
Assume 04 is the point with a rank 4 polar quadric and take the canonical form Vxy - Rzt = 0 (V, R -f 0) for this quadric [4).
W.l.o.g. assume 03 is the nucleus whose polar quadric has rank 3; then equations (1.2) become
266
M. de Finis and M.J. de Resmini
r - Exy2 Ex
)
(6.27)
Fx 2
1l
Fxz
+
Hy 2
- Hxy - Jyz 2 + Jy2 + Rt
Vxy - Rzt
+
Syz
+
Vyt = 0
+
Sxz
+
Vxt = 0
+
Sxy
=0 V, RIO, FJ - S2 I 0 .
=0 ,
Three cases must be considered [4]: 2 2 (i ) JV + RH2 I o and FV + E2R I 0 2 (i i) Jv2 + RH2 o and FV + E2R I 0 (or conversely) ; 2 (iii) JV 2 + RH2 o and FV + E2R = 0 (i) With suitable coordinates, S
= E = H = 0 can be assumed in equations
(6.27), while F, J, R, V I 0; then the four nuclei beside 0 are: 3 x = v'JR, y = v'FR, z = V, t = v'FJ; x = - v'JR, y = v'FR, z = V, t = -v'FJ; x = v'JR, Y = - v'FR, z = V, t = - v'FJ;
(6.28)
x
=
- v'JR, y
=
- v'FR, z
=
V, t =
v'FJ.
Since they are independent, F has five nuclei, and prop. 5.1(a) holds [4]. 2 2 2 2 (ii) Suppose JV + RH = 0 and FV + E RIO. Then a nucleus exists on the line x
= z = 0 and its polar quadric has rank 2; so, prop. 6.1(ii) holds. (iii) In this case another nucleus exists on the plane z = 0, thus, there
are two nuclei, both with a rank 2 polar quadric, and a non-degenerate conic all whose points are nuclei exists, and prop. 4.1 holds [4] (see also prop. 6.1).
7. ON CUBIC SURFACES WITH FIVE NUCLEI AND ON CUBIC SURFACES WITH FIVE NUCLEI AND 27 LINES It was already shown that cubic surfaces with five nuclei exist; now it will be proved that this is the general case. PROPOSITION 7.1: If a point exists whose polar quadria has rank 4, then the most general aubia surfaae has five nudei, no four of them aoplanar.
PROOF: Choosing coordinates as in the proof of prop. 6.26,
~
is defined by
267
On cubic surfaces over a field of characteristic 3
r -
(7.2)
)
"\
l
2 2 Exy - Fxz + Hy + Lz + Syz + Vyt El - Hxy - Jyz + Mz2 + Sxz + Vxt Fl + Jy 2 - Lxz - Myz + Rt 2 + Sxy Vxy - Rtz
=0 =0 =0
o,
V, R t-
o.
Under the assumptions HM2 + LJ 2 + SMJ
(7.3)
t-
0 and EL2 + MF2 + FLS
t-
0 ,
solving the simultaneous equations (7.2), the following is obtained [4]: x5(FLSV 3 + EL 2V3 + F2MV 3) + x4(FRS2V 2 _ FHLRV 2 - F2JRV 2 + L2V4 _ EFMRV 2) + x3(EFJR 2V + EHLR 2V + E2R2MV + FMRV 3 + LRSV 3 - ER 2S2V) + x2(_ FH2 R3_ E2 JR 3 _ FJR 2V2 + EMR 2V2 - EHR 3S) + x(MR 2V3 - HR3SV - EJR 3V) - JR3V2 - H2 R4 = 0 , t =1
(7.4)
Therefore, five (generally distinct) nuclei exist and by prop. 4.1 no four of them are coplanar. 2 When HM2 + LJ + SMJ = 0, the second condition in (7.3) being satisfied, 2 2 prop. 6.26 holds. When also EL + MF + FLS = 0, prop. 6.1 holds [4]. It is not difficult to verify that the surfaces for which E is given by (7.2)(and (7.3) hold) have 27 lines [4]. Cubic surfaces containing 27 lines have already been considered. Now it will be shown that also over a field of characteristic 3 a cubic surface containing 27 lines and having the same canonical form as in the classical case can be constructed; furthermore, such a surface has five distinct nuclei (no four of them being coplanar). Starting from equation (1.1), assume t = 0 is a tri-tangent plane intersecting F at the degenerate cubic curve xyz
= t = 0; then the equation of F
becomes: (7.5)
3222 222 Dt + Gx t + Ky t + Nz t + Pxt + Qyt + Rzt + Sxyz + Txzt + Vxyt + Wyzt
=0 •
= A z meeting F at a degenerate cubic curve consisting of three lines [4], assume one of them is z = 0 and the cubic curve on Since there are five planes t
it contains 04. So, D = 0 in eqn. (7.5) and 2 (7.6) PQV - p2K - Q G = 0
268
M. de Finis and M.J. de Resmini
(and the other three planes through t
= z = 0 intersecting F at lines are distinct
if F does not contain any of its nuclei (41). Bya similar argument (4), there are five planes through the line y meeting F at lines; assume one of them is y
=t =0
= 0; then
2 2 PRT - P N - R G = 0
(7.7)
must hold.
Also tht'ough x
t
=
° there are five planes intersecting F at lines;
=
take one of these planes as x = 0; then
(7.8) must hold. Taking into account (7.6), (7.7), and (7.8), being D (7.9)
G
K
0, eqn. (7.5) becomes
N
5xyz +t(pX + oy + RZ + t)(Px + Qy + Rz) = 0 ,
(5 f. 0).
This surface contains 27 lines and its equation is the classical one [1]. The cubic surface defined by equation (7.9) has five distinct nuclei (none of them on it), provided that G, K, N, P, Q, R f. 0[41. As special cases (for more details see (4)), since
(7.10) the following are obtained: (i) G = 0 and TV
P5 f.
°:
F
has a conic node (and four nuclei not on it).
(ii) G = P = 0 : F has a binode, the biplanes intersect on F; (7.9) becomes (7.11)
2 2 5xyz + t(Ky + Nz + Wyz + Qyt + Rzt) =
°;
the binode takes into account three nuclei, and two other nuclei exist (prop. 5.6 (l)(a) holds). (iii) G = 0, P f. 0, S
=
PNK OR
=
TV
p : F has a binode and in the most general
case the biplanes do not intersect on F (prop. 5.6(i)(b) holds). (iv) G = K = 0 (N f. 0), P, Q f.
° : F has two conic nodes and there are three
other nuclei (prop. 6.1 (i) holds).
°:
(v) G = K = 0, P = 0, Q f. F has a binode and a conic node (prop. 6.1(ii) holds). (When also Q = 0, F splits into a plane and a quadric.) (vi) G = K = N = 0, P, Q, R f. 0 : F has three conic nodes and two other nuc 1ei not on it. (vii) G = K
N
P
0, Q, R f. 0
F has two conic nodes and a binode, the
On cubic surfaces over a field of characteristic 3
269
biplanes intersecting on F, and no other nucleus exists (prop. (5.6)(i)(a) holds). No other non-degenerate case can be obtained from equation (7.9).
8. ON CLASSIFICATION OF CUBIC SURFACES In this section the classification of cubic surfaces according to the numbers and types of singularities (i.e. the classification over the complex number field) will be compared with the classification based on possible configurations of nuclei, taking into account the loci of tri-tangent lines and the planes intersecting F at a line counted three times. In this comparison, neither cones nor ruled cubic surfaces will be considered.
(The former can be classified according to the classification of plane
cubic curves [6], the latter were completely characterized in sect 3.) First, recall some classical notation for singular pOints [1]: C2 B3
conic node;
B4
binode with biplanes intersecting on F;
B5
binode with biplanes intersecting on F at s, one biplane meeting F at s
binode with biplanes not intersecting on F;
counted twice and at another line; B6
binode with biplanes intersecting on F and one biplane meeting F at the line s counted three times;
U6 U 7
uninode such that the uniplane meets F at three distinct lines; uninode such that the uniplane meets F at two coincident (and one distinct) lines;
U ~ uninode such that the uniplane meets F at three coincident lines. 8 In the following table the 35 different types of cubic surface other than cones and ruled surfaces over an algebraically closed field of characteristic 3 are listed and the notation here below is used: C+ P
~
the locus of nuclei consists of a non-degenerate conic and a point not on its plane;
n(v)
number of points whose polar quadric vanishes;
n(n)
number of planes meeting F at a line counted three times.
270 #
M. de Finis and M.J. de Resmini
and type of nuclei
n(v)
singularities configuration
locus of
ref.
n(1I)
tri-tangents
5 distinct
0
5 cones
0
5.1(a)(i) & (7.9)
3 distinct
0
2 cones & 2 noncoplanar pencils
0
5.6(vi)
2 distinct
0
1 cone & 2 noncoplanar pencils
0
6.1(ii)2.(d)
1 only
0
1 penci 1
0
6.16.2.(i)(b)
5 distinct
0
4 cones
0
5.1 (a)(ii)
3 distinct
0
1 cone & 2 noncoplanar pencils
0
5.6( i v)
2 distinct
0
2 non-coplanar penci 1s
0
6.1(ii)2.(c)
1 line
0
special 1ine complex
0
3.1 (i )(b.2)
1 1i ne
0
all 1i nes through a point
3
2.5(b)(3ii),(2.23)
3 distinct
0
2 cones
0
5.6(v)(b)
2 distinct
0
1 cone
0
6.1 (i i )2. (b.2)
B4
3 distinct
0
2 cones
0
5.6(v)(a)
B5
2 distinct
0
1 cone
0
6.1(ii)2.(b.l)
B6
C+ P
0
1 cone &all lines on a plane
1
4.1(iii)
U6
1 line
0
special line complex
0
3.1(i)(b.l)
1 only
0
~
0
6.16,2(i)(a),(6.23)
U7
1 only
0
~
0
6.16,2(i )(a) ,(6.24)
Us
2 incident 1ines
0
all 1i nes on a plane
1
3.1(ii)(b)
1 1i ne
1
all lines through 3 a point
0
C2
B3
2. 5( b)( 3. iii)
On cubic surfaces over a field of characteristic 3
#
and type of nuclei
n(v)
singularities configuration
locus of
n(1f)
271
ref.
tri-tangents
5 distinct
0
3 cones
0
5.1(a)(iii)
3 distinct
0
2 non-coplanar pencils
0
5.6(ii)
3C 2
5 distinct
0
2 cones
0
5.1 (a)(iv)
4C 2
5 distinct
0
1 cone
0
5.1 (a)(v)
2B3
2 incident 1ines
0
all 1i nes on a plane
1
3.1 (ii )(a)
C+ P
0
1 cone and all lines on a plane
1
4.1 (i)
2 incident 1i nes
1
all 1i nes th rough a point
3
2.5(b)(2. ii)
3 concurrent non-coplanar 1ines
1
all 1i nes through a point
3
2.5(b)(1. ii)
3 distinct
0
1 cone
0
5.6(iii)(b)
2 distinct
0
~
0
6.1 (ii )2. (a.2)
B4 + C2
3 distinct
0
1 cone
0
5.6(iii)(a)
B5 + C2
2 distinct
0
~
0
6.1(ii)2.(a.l)
B6 + C2
C+ P
0
1
4.1(iv)
B3 + 2C 2
3 distinct
0
0
5.6(i )(b)
2B3 + C2
C+ P
0
1
4.1(ii)
B4 + 2C 2
3 distinct
0
0
5.6(i )(a)
2C 2
3B3
B3 + C2
-
(Remark.
all 1i nes on a plane ~
a 11 1i nes on a plane ~
Of course, only proper tri-tangent lines, i.e. not through any
singular point, were considered.)
M. de Finis and M.J. de Resmini
272
9. A COMBINATORIAL CONFIGURATION AND THE DESIGN ASSOCIATED WITH IT In this section only cubic surfaces having exactly five nuclei and no singular point will be considered. Any tri-tangent line through two nuclei will be called a special tri-tangent; thus, for such an F there are exactly 10 special tritangents; therefore 10 points on F are determined and in this set of points there are exactly 10 collinear triples and 5 coplanar hexads. The set conSisting of the 10 points on F belonging to its special tritangents, the 10 triples of collinear points (i.e. the 10 lines containing these triples) and the 5 planes containing the hexads will be called configuration N (N for short).
If (5.4) is the equation of F, then the 10 pOints in N are: 1
= (0,0,
3-
v D, - ~C)
3-
6 '"
3-
3-
3- 3- 3-
= (0, v D,O, - v B) 333 = (0, v c, - v B,O) 34 = (v D,O,O, - ~A) 35 = (v C,O, - ~A,O)
2
3-
(v B, - v A,O,O) 3
7 '" (v D, v D, v D, - v A + B + C + 1) 3- 3-
8 = (v
c, V C,
3-
3
3
3-
- v A + B + D + 1, v C) 3- 3-
9 '" (v B, - v A + C + D + 1, v B, v B) 10
= (-
3
3- 3- 3-
vB + C + D + 1, vA, vA, vA).
The tangent plane at each of these points intersects F in a cubic curve having a cusp at that point (the tangent being the special tri-tangent) and the polar quadric of any of these points contains the special tri-tangent through it. When A '" B = C = 0 = 1 the polar quadrics of all points in N have rank 2, one of the planes in which they split being tangent to F (the pairs of planes do not intersect at the special tri-tangents). The planes of N are: vAx+vBy
+VCz
3-
+ ~O t
=D
33'A : V A+l x + V By
3+v Cz
+ qO t
=
0
3-
+ qO t
=
0
=
0
'IT
3-
3-
"
3-
3"B ::;;Ax
+ v B+1 y + v C z 3-
3-
+vBy
3-
3-
'ITC :vAx
3-
3-
3-
3-
+ V C+ 1 z + V' D t
lTD :V'Ax + vB Y + v C z + V' D+ 1 t = 0 and the hexads are the following ones: 2
3
4
5
6
3
5
6
8
9
10
2
4
6
7
9 10
4
5
7
8
TI
"D TIC: '"B:
10
273
On cubic surfaces over a field of characteristic 3
2
3
7
8
9.
Finally, the lines of N (i.e. the lines containing the triples of collinear points) are: 1TD n 1TB
5
8
10
5
1TD n 1TA
3
8
9
1T n 1TA
2
3.
1T n 1TB
4
1T n 1TC
2
4
6
1TC n 1TB
4
7
10
1T n 1TD
3
5
6
1TC n 1TA
2
7
9
1T Dn 1TC
6
9 10
1TB n 1TA
7
8
Generally, the planes of N are not tri-tangent planes to
F',
however, in the
family of all cubic surfaces having the same five nuclei there is a surface
~~
F
to
which all planes in N are tri-tangent planes (but they are not the only tritangent planes).
i:
The equation to F is:
3 3 3 3 x + y + z + t + xyz + xyt + xzt + yzt
(9.1 )
=0
An easy but tedious computation shows [4) that the five tri-tangent planes ,~
of N contain exactly 15 lines belonging to F and these lines are the only lines on F": over the prime field of K. Now a design to describe Nwill be defined. First, remark that any hexad of coplanar points of N, taking as blocks the triples of collinear pOints, is a PBIBD with two association classes and parameters (v,b,r,k,A ,A ,n l ,n 2) = (6,4,2,3,1,0,4,1), two points being first associates l 2 if they belong to a block and second associates if they do not [2). The definition of a composed PBIBD (which up to the authors' knowledge is new) is the following one. A set of p PBIBD's with the same parameters (b,v,r,k,A ,A ,n ,n ) and two l 2 l 2 ~Iith parameters (p,B,r,b,A ,A , l 2 Nl ,N ) and two association classes if the set consisting of the B blocks of the p 2 given PBIBD's is a PBIBD (whose points are exactly the blocks of the p PBIBD's)
association classes is a CPBIBD (Composed PBIBD)
containing p blocks of size b, such that through any point there are r blocks. Consequently, the parameters of a CPBIBD must satisfy the following equalities: (9.2)
vr
= bk,
nl + n2
=v -
(9.3)
Br
= pb,
Nl + N2
= B-1,
1, nlA l + n2A2 N1Al + N2A2
= r(k
- 1) ,
= r(b
- 1)
274
M. de Finis and M.J. de R esmini
(Of course, starting from p PBIBD's with m association classes, and the same parameters, a CPBIBD with m association classes can be defined.) When
N
is considered, the p PBIBD's are the already defined 5 PBIBD's
consisting of the coplanar hexads; now, call points the blocks (triples of collinear points) of N, first associates two points on the same block (i .e. two coplanar triples of collinear points) and second associates two points through which no block exists (i.e. two non-coplanar triples of collinear points); thus, a CPBIBD with parameters p
5,B
10,b
4,r=2.A l =1,A 2 =0,N l
(v
6, k
3, ;'1
1, "2
= 0,
n
l
= 4,
n
2
6,N 2
3
= 1)
is obtained, and (9.2), (9.3) hold.
10. ON CUBIC SURFACES WITH FIVE NUCLEI IN PG(3,3) Usually, when cubic surfaces are investigated in PG(3,q), the assumption q >3 is made [5],
[7),
since lines on the surface are looked for. On the other
hand, the existence of configuration N motivates the investigation of cubic surfaces in PG(3,3). PROPOSITION 10.1: In PG(3,3) there are but two projectively inequivalent cubic surfaces having five nuclei, none of them on F.
The surfaces of the first type
contain exactly 25 points over GF(3); namely, the 10 points of N and the 15 points in PG(3.3) not belonging to any tine joining two nuclei.
The planes of N are the
t!'i-tangent planes to F and together they contain aU 15 points of F, 9 on each plane (two planes sharing exactly 3 points on F).
The lines of N do not belong
to F.
The cubic surfaces of the second type contain exactly 16 points; these points belong to three out of the five planes of N; these three planes have a unique common point and each of them intersects F at a line and an irreducible conic.
The three lines (one on each plane) are coplanar, their plane not being a
pZane of N, and have a conmon point. It; points on F.
Together these tines contain 10 out of the
The remaining 6 points be long to the three conics.
The planes
of these conics pairwise meet at secant lines of the conics; thus, two planes and the conics on them (which have two conmon points) determine the third plane (the
On cubic surfaces over a field of characteristic 3
latter being the plane through the unshared pairs of points). belong to the same elliptic quadric. just six points of N.
275
The three conics
Each of the other two planes of N contains
Therefore. F consists of the points on three plane
section~
by three planes through the same point p. of an elliptic quadric (the planes pairwise meeting at secant lines of the quadric) and of the points on three concurrent coplanar lines on a pZane through p which is tangent to the quadric at a point not in N.
PROOF: Since the proof consists of tedious computations and checking [4), here only the canonical forms for the surfaces will be given. Taking as nuclei the vertices of the reference simplex and the unit point, the equation of the 25point surface is ( 10.2)
3 3 3 3 x + Y + z + t + xyz + xyt + xzt + yzt
0,
while the equation of the 16-point surface is either (10.3)
3 x3 + y3 - z
- t 3 + xyz
3 3 x3 - y - z
- t 3 + xyz + xyt + xzt + yzt = 0 •
+
xyt + xzt + yzt = o
,
or (10.4 )
For the sake of completeness, the main results on cubic surfaces in PG(3,3) with five nuclei and singular points will now be stated (for more details see [ 4 J).
(i) Four of the five nuclei belong to F. All such surfaces are projectively equivalent and the equation of F (choosing coordinates as in prop. 10.1) is (10.5 )
xyz + xyt + xzt + yzt
=0 .
F has exactly 19 points and 9 lines; N degenerates, its points being the 4 conic nodes and its planes the 4 planes joining them by threes. (ii) Three of the five nuclei belong to F. All such surfaces have three conic nodes and are projectively equivalent to (10.6)
x3 + xyz + xyt + xzt + yzt = 0 ,
whose conic nodes are O2 , 0 , 0 , 3 4 F contains exactly 16 points and these together belong to two planes.
The
plane joining the three conic nodes contains 9 points (on the lines joining pairs
276
M. de Finis and M.J. de Resmini
of conic nodes); the other 7 points belong to another plane (namely, y
z
+
t
+
=O~
and their configuration is shown in the picture at the left.
(The two planes meet at a
line not belonging to F.l
N degenerates; its points are the three conic nodes and the point p on the unique special tri-tangent line (not through any singular point); its planes are the plane through the three singular points and the three planes projecting from p pairs of conic nodes. (iii) Two of the five nuclei belong to the surface and are conic nodes.
All
such surfaces are projectively equivalent to 3
3
x - y
(l0.7)
+
xyz
+
xyt
+
xzt
+
yzt
0,
having its conic nodes at the points 03 and 04' F has just 12 points, four of them on the line
t
through 03 and 04'
The
other points belong to two planes through a point p on t; these planes meet at a line tangent to F at p and intersecting F at q # p. through
t
Two of the four planes
contain two points of F not on t; the remaining two just one point on F.
N degenerates: there are just 5 points of N, but all five planes exist; the three points of N distinct from the conic nodes are collinear. (iv) The surface contains just one of its five nuclei. therefore this point is a conic node. inequivalent types of such surfaces.
Assume it is 04;
There are exactly two projectively The only possible equations of a cubic
surface wi th five nuclei, one of them on the surface, are (l0.8)
x3
(10.9)
3
x
(10.10)
+ +
y3 3
+
3
z
3
y - z
3 3 3 - x - y - z
+
xyz
+
xyt
+
xzt
+
+
xyz
+
xyt
+
xzt
+
+
xyz
+
xyt
+
xzt
+
°, yzt = ° yzt = ° yzt =
Equations (10.8) and (10.9) yield two projectively equivalent surfaces, while equation (10.10) yields another type. The surface (10.8) contains exactly 10 points. points coinciding in 04; there are only 4 planes.
N degenerates, four of its
The 9 points on F distinct
from its conic node belong to two planes; each of them contains a triple of collinear points - the same for both planes - and a triple of non-collinear One of these planes contains three of the five nuclei, the other none.
point~
(The same
On cubic surfaces over a field of characteristic 3
277
is true for the surface (10.9).) Also the surface (10.10) contains exactly 10 points, but the points distinct from its conic node are precisely the points on three non-concurrent coplanar lines, which are the only lines on the surface.
BI BLIOGRAPHY 1. 2. 3. 4.
5. 6. 7.
W.H. Blytbe, liOn modeZs of mAhia surfaaes", Cambridge U.P. 1905. R.C. Bose, Graphs and Designs, in "Finite Geometria Struatures and their AppZiaations", CIME (1972), Ed. Cremonese, Roma, 1973. W. Burau, AZgebraisahe Kurven und FZaeahen, Band II, W. De Gruyten and Co., Berl in 1962. M. de Finis and M.J. de Resmini, Sulle superficie cubiche in uno spazio proiettivo tridimensionale sopra un campo a caratteristica 3, Quad. Sem. Geom. Comb. n. 36, Maggio 1981, 1st. Mat. "G. Castelnuovo", Univ. Roma. J.W.P. Hirschfeld, Classical configurations over finite fields: I. The double-six and the cubic surface with 27 lines, Rend. Mat. e AppZ., 26 (1967), 115-152. J.W.P. Hirschfeld, ''Projeative Geometries over Finite FieZds", Clarendon Press, Oxford, 1979. J.W.P. Hirschfeld, Cubic surfaces whose points all lie on their 27 lines, in P.J. Cameron et al. (ed.), Finite Geometries and Designs, Cambridge 1981, pp. 169-171.
Istituto Matematico "G. Castelnuovo" Citta Universitaria 00185 Roma Italy
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Annals of Discrete Mathematics 18 (1983) 279-282 © North-Holland Publishing Company
279
SOME CHARACTERIZATIONS OF FINITE 3-DIMENSIONAL PROJECTIVE SPACES AND AFFINOPROJECTIVE PLANES A. Delandtsheer
A lineaP space is a non-empty set of elements called points together with a family of distinguished subsets called lines such that any two distinct points are contained in exactly one line, each line containing at least two points. A linear space is said finite if it has a finite number of points. A transversal of two lines Land L' will be any line intersecting LUL' in two distinct points. The generalized projective spaces may be defined as the linear spaces in which for any two disjoint lines Land L', any point p 4 LUL' is on at most one transversal of Land L' (the dimension being';; 3 iff "at most one" is replaced by "exactly one").
The finite affine planes of order n have a similar property: for
any two disjoint lines Land L', any point p 4 LUL' is on exactly n transversals of Land L
I.
It is natural to try to classify all finite linear spaces having such a property. THEOREt1 (A. Beutelspacher and A. Delandtsheer, [1]): Let S be a finite linear space.
If there is a non-negative integer t such that for any two disjoint lines
L, L of S, any point p outside L and L is on exactly t transversals of Land L 1
1
I,
then one of the following possibilities occurs: (i) S is a generalized projective space (moreover, if the dimension of S is
at least 4, any line has exactly two points)
(ii) S is an affine plane, an affine plane with one point at infinity, or a punctured projective plane
(iii) S is the Fano quasi-plane (i.e. the linear space obtained by breaking one line of PG(2,2) into three lines of two points).
Note that in case (ii), S is always an affino-projective plane, that is a linear space obtained by deleting any number m of collinear points from a projec-
280
A. Delandtsheer
tive plane of order n.
S is an affine plane if m = n+1, an affine plane with one
point at infinity if m = n, and a punctured projective plane if m = 1.
A similar problem is to classify the finite linear spaces satisfying the following condition (1,) for any two intersecting lines Land L', any point outside Land L' lies on
exactly t transversals of Land L'. PROPOSITION: If S is a finite linear space containing at least two lines and for JcJ,:/ic;: -:;her"! is all hlteger t such that ("') is satisfied, then
(i) S i3 a degenerate projective plane
(ii) S is a Steiner system S(2,k,v)(k;;' t+l)
such that
(1) ti(k-1)(k-2) (2) kil (k-1)(k-2)/t+21l (k-1)(k-2)/t+1J (3)
t(2k-2-t)ik(k-1)2(k-2)
hi particuZai>. S is a projective plane if t
k-l and S is an affine plane
Conditions (1) and (2) are obtained by easy counting arguments.
Condition
(3) is a consequence of the existence of a partial geometry (k,k-1,t) defined as follows: given a line L of S, the points of the partial geometry are the points of S which are not on L and the lines are the restrictions to S-L of the lines of S intersecting L. Brouwer has observed that a symmetric 2-design is obtained by taking two points p and q of S, by call ing "points" the lines through p distinct from the line pq and "blocks" the lines through q distinct from pq and by saying that a point and a block are incident iff the corresponding lines intersect.
If we apply
the Bruck-Ryser-Chow1a theorem to this symmetric 2-design St(2,k-1,(k-1)(k-2)/t+1) we get a fourth condition on the parameters t and k. These four conditions together rule out many values of t and k but there remain infinitely many values which satisfy all those conditions, for example t
= gn and k = (t+1)(t+2)/2. In order to get not only affine and projective planes but also affine and
projective spaces of any dimension, it seems natural to replace in ("') "t trans-
Characterizations of finite 3-spaces and planes
versals" by "0 or t transversals".
281
There is no hope that this condition charac-
terizes affine and projective spaces.
However, it is possible to prove:
PROPOSITION: Let S be a finite linear spaae suah that (a) for any two interseating lines Land L', the set of points which are on
L U L' or on at least one transversal of Land L' is a linear subspaae of S (b) there is an integer t suah that for any two intersecting lines Land L', any point outside L and L' is on exaatly 0 or t transversals of L and L' Then
(i) if t
~
2, S is a planar spaae all of whose lines have the same number
of points and in which all planes satisfy condition (*) In partiaular, if t = k-l S is a projeative space,if t = k-2 S is an affine spaae
(ii) if t = 1, S aonsists of disjoint pieaes joined by lines of two points, these pieces being - planar S(2,3,v.) all of whose planes are affine 1
- lines having more than two points - points
(iii) if t
0, S is a S(2,2,v).
Another similar problem is to classify the finite linear spaces for which there is a positive integer s such that for any two disjoint lines Land L', any point outside Land L' is on exactly s lines disjoint from both Land L'.
THEOREM: If S is a finite linear space for whiah there is a positive integer s such that for any two disjoint lines Land L', any point outside Land L' is on exaatly s lines disjoint from both L and L', then one of the following oaaurs:
(i) S is a projeative spaae of dimension d
~
3
(ii) S is a degenerate projeative plane, a generalized projeative spaae aonsisting of two disjoint lines of the same size or a trivial linear spaae
S(2,2,v) (iii) S is an affino-projeative plane (but not an affine plane with one point at infinity)
(iv) S is an affine plane from whiah either one point or one line has been deleted.
The proofs of these theorems will be published elsewhere.
282
A. Delandtsheer
BI BLIOGRAPHY
1.
A. Beutelspacher and A. Delandtsheer, A common characterization of finite projective spaces and planes, European JOUPnal of Combinatories (1981) 2, 213-219.
Universite Libre de Bruxe11es Campus Plaine, C.P. 210 Boulevard du Triomphe, B - 1050 Bruxelles Belgium
Annals of Discrete Mathematics 18 (1983) 283-290 © North-Holland Publishing Company
283
ON A PROBLEr~ OF A. KOTZIG
J. Denes
In 1975 during the Graph Theory Conference in Oberwolfach, Prof. A. Kotzig suggested to me the following problem: Let Q be a finite quasigroup, a graph TQ corresponds to Q in the following way: The vertices of TQ are labelled by the elements of Q and an edge is directed from a to b (a,b E Q) if and only if b=a -1 l l where a- denotes the local right inverse of a, i.e. ae = a, aaea' What a kind of properties of Q can be characterized by T ? Q Obviously TQ is a graph which has been called a transformation graph, see [31.
A regular transformation graph is called a permutation graph.
In several
papers transformation graphs have been studied in detail and all our results can be found in [41.
For definitions of loop, isotopy et cetera the reader is
referred to [ 51. In 1976 a computer search was set up.
The graphs corresponding to some
quasi groups of order 6 (which are the class representatives, for definitions see
[ 51) were determined. These graphs will be published in [41. No definitive solution was obtained by the computer investigation.
In the
meantime we had correspondence with Dr. A.D. Keedwell and that inspired the present author to prove the following: THEOREM 1: Let Q be a finite loop of order n, then there exist isotopiaquasigroups Q=Ql,Q2, ••• ,Q(n-l).1 with right identity element suah that TQ ,TQ , ... TQ 1 2 (n-l)! are different permutation graphs with n labelled vertiaes. (The isotopes are obtained using (n-l)! aolumn permutations).
PROOF: Let us consider a latin square which is the multiplication table of loop Ql of order n. The isotopes Q2 ••• Q(n-l)! of Ql will be quasi groups with the same right identity element if we permute n-l columns and fix the column where the border
el~
284
J. Denes
ment is the right identity element.
(See Figure 1. where the case when n=3 is
Let Li i=2.3 ••••• (n-1)! denote the multiplication table of 0;.
exhibited).
If
the cells which are occupied by the right identity element are replaced by 1 and all others by O. then to Li there corresponds a permutation matrix P.1 and P1'P 2 •••• 'P(n-1)! are all different. It is easy to see that Pi is the adjacency matrix of TO. and this completes the proof. 1
'1
2
3
2
3
0
0
1- 1=1. 2- 1=3. 3- 1=2
2231---+00 3
2
3
o
o
o
() 3
2 3 3
o2 =
2
2
2 3 --
0
o
0
o
332
0
0
123 T
O2
000
=
Figure 1. REMARK: It is easy to find an example of a quasi group 0 such that TO is not a permutation graph (see Figure 2.).
I
1 2 3 '
2 3
0
2 3
2 3
T • Q
3
2
Figure 2.
V
3
285
On a problem of A. Kotzig
As we have seen by theorem 1. isotopism does not keep the Kotzig graph of a quasigroup. Theorem 2 will show us that this is not the case if isomorphism is considered instead of isotopism. THEOREM 2: Let Q and Q2 be two isomorphic quasigroups, then TQ and TQ are isol 1 2 morphic graphs, (i.e. graphs which are equal, apart from labelling).
On the con-
trary if Q and Q are not isomorphic, TQ and TQ might be isomorphic.
l
PROOF: Clearly if
2
1
2
denotes the isomorphism between Ql and Q2 then for every a E Ql' ~(a) = ~(a e ) = ~(a) ~(e ) holds. This implies that e ( ) = ~(e ) -1 a_ l -1 -1 cp a a a = ~(a a ) = ~(a) cp(a ) as a consequence ~(a ) = cp(a) holds and TQ and TQ ~
1
2
are isomorphic.
It is obvious that TQ and TQ are isomorphic if Q and Q2 are l 2 two non isomorphic groups of order n ifl n is a composite odd integer. and this
completes the proof. Theorem 3 will show us that there are other types of TQ graphes than implied by theorem 1. To formulate theorem 3 we shall need some definitions. (see also [51): a transversal of a latin square of order n is a set of n cells one in each row. one in each column. and such that no two of the cells contain the same symbol. A transversal is called cyclic if it has the form
THEOREt·l 3: If n is odd, n
~ 0
mod 3 then there exists at least one quasigroup Q
of order n such that TQ is a single cycle.
PROOF: For n odd and n ~ 0 mod 3 such a quasigroup (Q •. ) always exist. h.j = 3h-2j+4(mod n).
Then h·(h+l)
a
We define
h+2 for h = 0.1 •...• n-l defines a cyclic -1
transversal and we have h·(h+2)=h. so eh = h+2 and h
= h+l.
For fUrther results on cyclic transversals see [51. especially Theorem 9.1 .7. of [ 51.
It is easy to show that there exist quasi groups whose Kotzig graph is different from the ones given in Theorem 1 and 3. One can use (to obtain such
286
J. Denes
quasigroups) some results of K. Heinrich see
(7).
(9).
In [7) K. Heinrich gave a
partial solution of a problem of L. Fuchs who has posed the following problem (see [5J problem 1.9.): If n is any positive integer and n=n +n +••• +n
any fixed parl 2 k tition of n. is it possible to find a quasigroup Q of order n which contains sub-
quasigroups Ql' Q2'···.Qk of orders nl , n2 •• ··, nk respectively. whose set theoretical union is Q? K. Heinrich gave necessary and sufficient condition for the affirmative reply in the case when k
~
which is not necessary in the case k > 4. the same subject very recently see (9).
4.
She gave a sufficient condition
Also she gave some further results on In [9] she wrote the following: "Denote
by RP(a,b;s,t), a < b, a latin square of order n=as+bt with s disjoint subsquares of order a and t of order b.
We show that RP(a,b;s,t) exist for all s
that RP(a,b;l,t) exist if and only if t s
~
3 and (5-1) a
that for all t
~
~
~
~
3, t
~
3;
3; that RP(a.b;s.l) exist if and only if
b; and that there exist integers T=T(a.b) and S=S(a.b) such
T and s
~
S, RP(a.b;2,t) and RP(a.b;s.2) exist".
It is worth remarking that the fact that a latin square has no kxk latin subsquares is not equivalent to the corresponding quasi group having no subquasigroups of order k.
While the quasi group corresponding to a latin square with no
kxk latin subsquare has no subquasigroup of order k, it is quite easy to construct quasigl'oups having no subquasigroups of order k whose multipl ication tables contain kxk latin subsquares. In (5) (p. 486) there is a problem which asks for all sufficiently large n there exist quasi groups of order n which contain no proper subquasigroups.
This
problem (which has a trivial affirmative solution see (12). [17] and (18). has been attributed erroneously to A.J.W. Hilton). wanted to be proved.
In [10] one can find what Hilton
He wrote: "It is probably true for most orders n there is a
latin square with no proper sub-latin square of order greater than 1; however, Dr. A.D. Keedwell has pOinted out to me that this is not true for n=4.
A partial
solution of Hilton's problem can be found in [8) where K. Heinrich proved that for orders n=pq where p and q are distinct primes and n
F 6. there exist latin squares
which contain no proper latin subsquare. A restricted version of Hilton's problem has been suggested and investigated in 113). Namely is it true that for every n there exists a latin square L with no latin subsquare of order 2? The reply is in the affirmative, if n F 2,4. \61,
(7),
[13], [14J, 115}. [16J).
(see
There exists an obvious connection between
latin squares without subsquares of order 2 and the Kotzig graph.
If L represents
287
On a problem of A. Kotzig
the multiplication table of a quasigroup 0, then the necessary and sufficient condition for L have no latin subsquare of order 2 is that every isotope of 0 is without subquasigroup of order 2.
With the aid of Theorem lone can prove Theorem
4. THEOREr~
4: If L denotes a Zoop of order n. then there exist quasigroups with Zeft
identity isotopic to L such that they have no proper subquasigroup.
PROOF: A sufficient condition for a quasi group 0 not to have proper quasigroup, is that TO has a cycle of length m(m >[ a
E
%]).
To prove this let us observe that if
0 is contained in a subquasigroup so does its left inverse. Further it is
well known: if n is any positive integer then an upper bound for the order of a subquasigroup of a quasigroup of order n is
[%]
(the greatest integer in
%),
(see
e.g. Lemma 8 of [18] and Theorem 1.5.3. of [5]). By the proof of our Theorem 1 it is immediate that for any given loop L of order n there exist isotopic quasigroups such that their Kotzig graphs contain cycles of length m(m > [~]). REMARK: One can use Theorem 3 instead of Theorem 1 to prove Theorem 4. If R denotes quasigroup of order n such that its multiplication table is a latin square which has no proper sub-latinsquare then every quasigroup isotopic to R is such that it has no proper subquasigroup. A sufficient condition for the latter property is that TRW
(R* denotes an arbitrary isotope of R) contains a
n
cycle of length m(m > [2]). This sufficient condition never fullfils, this is implied by Theorem 1. A groupoid G is said to have the right inverse property if, for each x in G, there is at least one b such that (xy) b=y for every y in G. The left inverse property can be defined analogously.
When a groupoid G has the right and the left
inverse property, then is said to have the inverse property. In the case of quasi groups the definitions above can be used too.
The read-
er can find a detailed study of quasi groups and loops with the inverse property (I.P. quasigroups) in [18],
[l~.
The best known I.P. quasi groups other than
groups are those characterized by the mild associative law a(b.cb) = (ab.c)b and the existence of a unique two sided identity element.
Finite quasigroups of this
type were introduced by R. Moufang by whose name one usually refers to them. A
288
J. Dl!nes
Moufang loop of odd order satisfies both Lagrange's theorem and Cauchy's theorem (see e.g. (5) p. 63}.
Consequently, the orders of all its elements and subloops
divide the order of the loops. A loop which satisfies the identity [(ab).c] .b=a[ (bc).b] is called a
BoZ
loor and such a loop satisfies the weak form of Lagrange theorem (namely that the
order of every element divides the order of the loop) (see e.g. [5) p. 32).
Since
Bol loops have the one sided inverse property, every Moufang loop is also a Bol loop. It is easy to see that if Q is a quasigroup with the right inverse property then TQ is a permutation graph.
Since every loop isotope of a Moufang quasigroup
is Moufang (see e.g. Corollary of Theorem 9 in [18J). if L is a loop isotope
R. Artzy independently from A. Kotzig defined and investigated the graph which exhibits the mapping which maps every elequasigroup, then TL is a permutation graph.
ment of a loop into its right inverse (see
[1
J).
Since he investigated loops with
one sided inverse property (he called them weak-inverse (WI) loops) his graphs are permutation graphs.
(It is a special case of the Kotzig graph).
Artzy proved
that if L is WI loop then \ consists of one sling (cycle of length 1) and m cycles of length k, such that k divides 2m, (see [11, [21).
WI loops which are
not Moufang are scare in the literature. One can find construction of non-Moufang WI loops in [ 11). Another special class of loops which have been studied by Artzy is the crossed inverse (Cl) loop.
Here we assume that to each x in G there corre-
spond a, b in G such that (xy)a=y=b(yx) for every y in G.
Artzy proved that a
necessary condition for a WI loop L not to be CI that TL consists a single cycle of length 1 and m cycles of length k such that k is a divisor of 2m. We would like to suggest a problem which can be consirlered as a semi group theoretical analogue of Kotzig's problem.
Let S be a finite semi group.
Since
every abstract semigroup of order n can be represented isomorphically as an appropriate subsemigroup of the symmetric semigroup of degree n+1 (F n+ 1)' without loss of generality we may suppose that S is a transformation semigroup. Every element of S could be represented as a transformation graph.
The restricted transforma-
tions to the elements in the cycles of the transformation graph will be called main permutation (really the graph representation of the main permutation is a permutation graph).
The main permutation of
a E
Fn will be denoted by f{a}.
1 2 3 4 5 6)
example, if a = ( 2 2 3 5 6 5
its graph representation is
For
On a problem of A. Kotzig
1
289
3
~O
Since the main permutation of is f(a)
_ (2 356) ,. ts 236 5
,,'ph "p""ot,t;oo ;, "follow,
c3
~
\5 J
6
The quasi inverse of a is the least power a k such that f(a)-1
k = f(a).
This notion
has been introduced in [3] and for further results see [4]. PROBLEM: Let S be a finite semigroup of order n, a graph TS corresponds to S in the following way; the vertices are labelled by the elements of S and an edge directed from a to b (a,b
E
S) if and only if b is the quasiinverse of a.
What
kind of properties of S can be characterized by TS? REMARK: Clearly TS is a transformation graph, but no further results could be obtained by the author. My sincere thanks to Mr. P. Ablonczy, Mr. B. Biro, Dr. A.D. Keedwell who kindly read the manuscript and made many helpful comments.
BI BLIOGRAPHY 1. 2. 3.
4. 5. 6. 7.
R. Artzy, Inverse cycles in weak inverse loops, Froc. Am. Math. Soc., 68 (1978), 132-134. R. Artzy, A functional equation solved by loops. Lecture given at the Functional Equations Conference at Graz, September 1978. For summary see Aequationes r4ath. 19 (1979), J. Denes, Connections between transformation semigroups and graphs, Theorie des graphes. Journee internationales d'etude. Rome, Juillet 1966. Dunod, Paris, Gordon and Breach, New York, 1967, 93-101. J. Denes, Transformations and transformation semi groups. To appear. J. Denes and A.D. Keedwell, Latin squares and their applications. Akademiai Kiado Budapest, Academic Press New York, English Universities Press, London 1974. R.H.F. Denniston, Remarks on Latin squares with no subsquares of order two, Utilitas Math., 13 (1978), 299-302. K. Heinrich, Latin squares composed of four disjoint subsquares, Combinatorial mathematics V. (Proc. Fifth Austral. Conf. Roy. Melbourne Inst. Tech., Melbourne 1976). pp. 118-127. Lecture Notes in Math. Vol. 622. Springer, Berlin, 1977.
255.
290
8. 9. 10.
11. 12. 13. 14. 15. 16. 17. 18. 10
J. Denes
K. Heinrich, Latin squares with no proper subsquares. Journal of Combinatorial Iheory A. 29 (1980), 346-353. K. Heinrich, Disjoint subquasigroups. To appear in Proc. London Math. Soc. A.J.W. Hilton, On the Szamkolowicz-Doyen classification of Steiner triple systems, Proc. London Math. Soc., 34 (1977), 102-116. K.W. Johnson and B.L. Sharma, Construction of weak inverse property loops, Rocky Mountain Journal of Mathematics, 11 (1981), 1-8. T. Kepka, A note on simple quasigroups, Acta Univ. Carolin. - Math. Phys., 19 (1978) no. 2, 59-60. A. Kotzig, C.C. Lindner and A. Rosa, Latin squares with no subsquares of order two and disjoint Steiner triple systems, Utilitas Math., 7 (1975), 287-294. A. Kotzig and J. Turgeon, On certain constructions for latin squares with no latin subsquares of order two, Discrete Mathematics, 16 (1976), 263270. M. Mc. Leish, On the existence of latin squares with no subsquares of order two, Utilitas Math., 8 (1975), 41-53. M. Mc. Leish, A general construction of latin squares with no subsquares of order two, Ars Combinatoria, 10 (1980), 179-186. N.S. Mendelsohn, Letter to the author dated October 12, 1976. R.H. Bruck, Some results in the theory of quasigroups, Trans. Amer. Math. Soc., 55 (1944), 19-52. R.H. Bruck, Contributions to the theory of loops, Trans. Amer. Math. Soc., 60 (1946), 245-354.
Csaba utca 10. 1122 Budapest Hungary
291
Annals of Discrete Mathematics 18 (1983) 291-294 © North-Holland Publishing Company
A SMALL 4-DESIGN R.H.F. Denniston
What is exhibited here is a t-design (repeated blocks not allowed) for which t=4,
k=5, v=12, A=4.
Infinitely many 4-designs are known, and a single one would
not as a rule deserve to be published; but this construction is so easy to grasp intuitively that is seemed to be worth talking about at the conference.
If a 4-
design with these parameters could be extended to a 6-design, that would be more interesting - but in fact this particular one cannot be extended. Our set of 12 points is divided into two hexads, which are represented in the tables as above and below a line. the hexads; where one
x
There is a one-one correspondence between
appears directly above another, it signifies that two
corresponding points belong to the subset with which we are concerned.
Each
hexad is unordered, but has a structure imposed on it which may conveniently be described in the language of Sylvester [2].
He used the term duad for an unor-
dered pair, and syntheme for a partition of the hexad into three duads; and he observed that a famity of five synthemes can be found which partitions the collection of 15 duads.
So, on our upper hexad, let us choose and fix one family (out
of the collection of six families that in fact exists): and let us use the one-one correspondence to carryover that structure to the lower hexad.
If we describe
two duads as "parallel" when they belong to the same syntheme of the chosen family, we have an axiom of a familiar kind:- Given a duad and a point not in it, we can find just one parallel duad that includes the point. We can now partition the whole collection of 5-subsets of our 12-set, as shown in Table 1, into two subcollections Land R.
This is done mainly by asking
how many pOints of a given 5-subset have been chosen from the upper hexad, and how many pairs of corresponding points are contained it it.
If, for instance, there
are two upper points and two pairs, the 5-subset goes into the subcollection Rand is said to be of type R4 ; two upper points, with no pair, give a 5-subset of type L . When, however, we come to two upper points with one pair, we need to discrim7
292
R.H.F. Denniston
inate by means of our fixed family of synthemes.
Namely, if the duad of upper
pOints and the duad of non-corresponding lower points are parallel, we have a 5subset of type L5 , otherwise of type R . Table 1 puts this in evidence by means 6 of vertical lines separating the duads of one syntheme. We proceed to show that, of the eight 5-subsets in which any given 4-subset is contained, four will be found in L and four in R.
In fact, the details, for
all the types of 4-subset that we can distinguish, are exhibited in Table 2. Discrimination between types L5 and R6 is easy in the second case on the right of Table 2. where the 4-subset includes a single upper point and also includes the corresponding lower point.
Namely, considering the two non-corresponding lower
points as a duad, we assert that just one parallel duad includes the single upper point, and therefore that our 4-subset can be embedded, in just one way, in a 5subset of type L5. In the next case down, where the single upper point has its corresponding lower point outside the 4-subset, we look at the duad of upper points with the property that the subset excludes them and also excludes their images below.
For
a 5-subset of type L , such a duad exists, and is parallel to the duad of chosen 5 upper points - but not parallel for type R6. And so we can make a similar assertion to the one at the end of the last paragraph. The other cases in Table 2 are easy to establish. that the ccllection L of 5-subsets
And the conclusion is
(and likewise, of course, the collection R) is
a 4-design lJi ttl the I'equiY'ed parameteY's.
The question remains whether we can be sure that L is not a contraction of a 5-design (k=6. v=13. A=4).
We can, however, apply a theorem proved by Alltop [1.
p. 184] about a (hypothetical) 5-design (Q,O) with these parameters, a point a of the point set
and a block
Q,
~
the six 5-subsets of the 6-set
(not including a) of the block set ~,
o.
Namely, of
either two or three are blocks of the contrac-
tion with respect to a, a 4-design which Alltop denotes by (Q ,Oa). a
But now (I go on from his theorem) suppose 6 is a 6-subset of not a block of O.
Q
a
which is
Then we observe that the complement C of 0 (in the collection
of all 6-subsets of Q) is another 5-design with the same parameters, and the a contraction C of C with respect to a is the complement of Va in the collection of 5-subsets of of
6,
Applying the theorem to C, we have that, of the six 5-subsets either two or three belong to Ca , and either four or three to Oa. Q .
a
293
A small 4-design
TABLE 1 L
R
x x x x x x x x x x
LZ
x
•
RZ
x x x x
x x x x
x • •
x x x x
• x
x
x x x x
x x x
x x
x x
x x x
RS
LS LS
x x x
x x
x x x
x x
TABLE 2 x x x x x x x x x x x x
2Ll. 2L3. 4RZ
x x x x
4LZ. 2Rl. 2R3
2Ll+. 2LS. 3Rz. IRs
x • x x x
3Lz. lLS. 2Rl+. 2Rs
2L3. 2L6. lRz. IRs. 2R7
• x x x x
ILz, IL S' 2L 7• 2R 3• 2R6
x X x x
4LlIJ 4Rl+
x x x x
ILl+. lLS. 2L 6 • 1RlIJ IRs. 2RS
t
x xl' • x x •
2LS. 2L7. 2RS. 2R7
x x I' '1' • x • x
2L6 • 2L7. 2R6. 2R7
294
R.H.F. Denniston
So we have a corollary to Alltop's theorem.
Suppose, if possible, that a
gl:ven 4-design (k=5, v=12, 1.=4) is a eontX'aetion of a S-design.
Then the nwnbeX'
of bloeksaontained in any given hexad must be two, thX'ee, oX' foUX'.
Since our L
in Table 1 includes all six 5-subsets of the upper hexad (and R of the lower), this condition is not satisfied.
BIBLIOGRAPHY 1. 2.
W.O. Alltop, Extending t-designs, J. CombinatoX'ial TheoX'y (A) 18 (1975), 177-186. J.J. Sylvester, Note on the historical origin of the unsymmetrical sixvalued function of six letters, London, EdinbuX'gh and Dublin Philos. Mag. and J. Sei. (21) (1861), 369-377.
Department of Mathematics University of Leicester Leicester LE1 7RH, U.K.
295
Annals of Discrete Mathematics 18 (1983) 295-314
© North-Holland Publishing Company
SOME NEW GENERALIZATIONS OF SHARPLY t-TRANSITIVE GROUPS AND SETS M. Deza
Recently the author tried to obtain permutation analogs of some of his previous results on largest set-intersection systems.
This gave rise to a series of
joint works dealing with the three following new concepts: a) permutation geometry (an analog of a matroid) which generalizes sharply t-transitive sets and some Jordan groups; b) (L,n)-sharp groups which generalize sharply t-transitive groups (the case L
{O,1,2, •.• ,t-l}); c) sharply edge-transitive digraphs which approximate a finite projective
plane over near-field, i.e. sharply t-transitive groups. This paper is a short up-to-date survey of results in this direction (without proofs but with some new observations and problems).
1. SET INTERSECTION MOTIVATION At first, I wanted to find a permutation analog of the following THEOREM 1.1 ([6), (7): Let A = A(L,k,v) be a family of k-subsets A. (i.e. 1
IA.I ,
=
k) ofa given v-set V such that any IA., () A.I J
E
L (A.,
~
A.) where J
= P o, 2 , ••• ,2r- l}' 0';; 20 < 21 < ... < Jl r- 1 < k. Then there exists a largest 1 nwnber Vo (L,k) (depending only on L and k) such that v> Vo (L,k) impZies: L
a)
([
6]) I A(L,k,v) I .;;
v - Jl. 11 - - ' ,
,
k - R..
b) ([ 7]) the equality holds above if and only if A(L,k,v) is the famiZy of hyperplanes of a perfect matroid-design (i.e. a matroid such that aZl flats of same rank have same cardinality); (see [11) and references to it for perfect matroid-designs).
296
M. Deza
The three most interesting (and most studied special cases of A(L,k,v) are: 1) L = {O,l, ... ,t-l}, a code with all distances dH(A.jA.) = IA. 1
- 2t
+
is a
S~;eineY'
J
1
l!.
A.I > 2kJ
lor, in other words, t-packing; the corresponding perfect matroid-design system 5(t,k,v) (which is also the smallest t-covering);
2) L = ~t,t+l , ... ,k-l },an anticode, considered by the famous Erdos-Ko-Rado's theorem; 3}
L = It}, (r,A)-design (r
= k,A
t) or, in other words, equidistant code
or weak D.-system. The first series of papers on permutation analogs concerned analogs of the three cases above : namely [8] for the case 1); [8] and [12] for 2); and [9] and most of its references for the case 3). The permutation analog of A(L,k,v) will be the set B = B(L,n) of permuta,
-1
tlons b.1
E
5n such that any f(b.1
o ~ £o
3. All sharply 2 (corresp. 3) transitive groups are known and they are exist iff n = pa (corresp. n = pa+l ).
The regular (i.e. sharply l-transitive group) L corresponds
exactly to a Latin square for which the rows form a group. Any cyclic Latin square and, for example, the Klein four-group provide examples but not all such Latin squares are known.
(Sharply t-transitive sets which are not groups we know
for t=l (any n), t = 2 (n = pal t
3 (n = pa + 1); but for larger t the existence
a
of such sets is an open problem).
It was proved in [13] that in the general case
I) we have IF(G)I = i and so G is exactly sharply t-transitive on the remaining n-ipoints of
fl.
In the case II (L = {i}) we know that G has a representation as a ({i},n)sharp group,i > 0, iff G has a G-invariant proper partition (and all such nonsolvable groups were classified).
For i = 1 we have IF(G) I = 1 and so G is
regular on the remaining n-l points.
The case i=2 and i=3 are classified in [14] •
•
For example, all ({2}.n)-sharp groups are A4 , S4. A , the generalized dihedral 5 group and. of course. the case IF(G)I = 2 with G regular on the n-2 remaining points. Let us consider the case III.
For the subcase L = {0.2} the classification
(see [17) is: D8 (n=4). Sn (n=6 with two different representations). GL(2.3) (n=8) and PSL(2.7) (n=14); moreover. G is transitive of rank 3 and G is a geometric group (see
§
4 for the definition).
(In [lJ, [3J many examples of ({O.l }.n)-
sharp groups. geometric or not are given; they are given implicitly in
§§
4-5).
The crucial reduction lemma in [13] says that for any ({i.£+s}.n)-sharp group IF(G) I ;;.
i-l+s~-(s-l)sl. where s,= max (1.[ (s-1)/21).
302
M. Deza
and the equality occurs for s s ;. 5).
= 1,2,3,4 (it is conjectured to be impossible for
Without lost of generality we can suppose (until the end of this section) that F(G) a)
=
0.
The classification of (U,£+s},n)-sharp groups given in
for L
{£,£~2}:
is:
either L = {0,2} (those 5 groups are given above) or
L = {1,3}, G has two orbits (n
=~
U
~I) and G ~,S4 (G~ = S3' G~I = S4' n=7) or
G;?; PSL(2,7) (G~ is 2-transitive of degree 7, G~ n a
[13]
is 2 transitive of degree 8,
15); b) for L = {£,
£~3}
: either L = {O,3} (and n = 6,9,15,24,27) or L = {2,5},
G has 3 orbits and G ;?; (71 3 x Z3) )() Z2 (n = 8). The 5 corresponding ({0,3},n)-sharp groups are: (Z3 x Z3 (Z3 x Z3)
~
~
Z2 (n = 6),
S7 (it has two different representations on 9 points), Z3 x PSL(2,4),
Z3 x PSL(2,7), and (Z3 x Z3 x Z3)
~
S4'
4. PERMUTATION GEOMETRY AND GEOMETRIC GROUPS Let us consider the set (denoted by Sin) of all bijections (in other words, partial permutations, subpermutations) between subsets of the set N = {1,2, ••• ,n}. Si we will denote by ran a its range and by a= dom a its n domain; the inverse a- l of a is the inverse map from ran a to dom a. The com-
For any bijection a
E
position aob of bijections a,b is defined (on its doman a~) by aob(i) = a(b(i)). The set Si with this composition aob (generalizing the composition of permutan
tions) becames an inverse semigroup.
It is called the symmetric inverse semi-
group and any inverse semigroup can be embedded (up to isomorphism) in it. Let a be a bijection (for example a = fl,3,4}
~
{2,3,1}).
We will
represent a in following ways: 1) the vector form, by the vector; (for example,
(a(l), a(2), ..• ,a(n)) with a(i) = 0 if
¢
a
a = (2,0,3,1));
2) the matrix form, by (O,l)-matrix «a .. ))n with a .. =1 if and only if a(i)=j (for lJ
n
lJ
example,
o1
0 0
000 0 001 0 1 000
);
3) in subset form; by the lal-subset A of the n2-set NxN of all ordered pairs
303
New generalizations of sharply t-transitive groups
(i,j) of integers I
~
i,j
~
n, with (i,j) E A if and only if a(i)
a
j (for
exampl~
A= {{1,2}, {3,3}, {4,l}}).
For any two bijections a,b we define now the new operation, meet C = a A b. In vector form, c(i)
=
b(i) if a(i)
=
b(i) and c(i)
=
0 otherwise.
In matrix form,
all Co 0 = min (ao o,bo 0) = ao 0 bo 0' i.e. «co 0)) is Hadamard product of matrices lJ lJ lJ lJ lJ lJ corresponding to a and b. In subset form C = A n B. The set Si with respect of n
this operation becames a semilattice (i.e. commutative idempotent semigroup). Now we define the new partial operation join c
= a v b for some a, b E Si n .
In matrix form, all Co 0 = max (ao o,bo 0) and c is defined if and only if «co 0)) is lJ lJ lJ lJ the matrix form of some bijection, i.e. if any column and any row of «co 0)) has lJ at most one element 1; in subset form C = AU B. We obtain partial the semilattice (Si n ,V) and the partial bisemilattice (S; n ,AlV), let us denote by the bijection (0,0, ..• ,0) (in vector form), i.e. with We call two bijections a,b disjoint if a A b = and
empty range and domain.
a v b is defined (if, moreover, a vb E S , we denote it by a
1
bo; also denote by
allc parallelism, i.e. the existence of b such that alb, c
1
b).
n
bijection a with
I§I= 1, i.e. of a form {I}
~
J
let us consider the following partial order a
~
b on Sin'
We say that a
(i.e. a covered by b) if a is the restriction of b to a subset of subset form, a
~
b means that A ~ B.
We have a
We call any
{j} a point (or singleton).
~
6a
~
b
dom bo; in J
b if and only if a A b
= a and
so the semilattice (Sin,A) with this partial order is a meet-semi lattice. (Actually, the partial bisemilattice (Sin,A,v) is meet distributive, i.e. the meet operation A distributes over the join partial operation v). The partial order a
~
b is a natural partial order in the inverse semigroup (also a-l';;;b- l );
(Si ,0), i.e., a ~ b implies coa ~ cob, aoc ~ boc for any c E Si n
we have a';;; b if and only if a
n
= eob for some idempotent e in (Si n ,0).
Now using these constructions A, V, .;;; on Si n we can define a permutation geometry; denote it by Perm G (l,n). let G be a subset of S ; denote by T(G) the n
meet-semi lattice of bijections generated by G; i.e. T(G) o
= {a A b:
a,b E 6}.
let
-1
b): a,bE G, a -; b} = l = {~'~l'''''~ o r- I}' (In the special case when G is a group l = {f(a): o ~ ~ o < ~l < ... < ~ r- I < n. a E G, a -; < I >} holds, i.e. G is a group of type l). It is evident that for any G be a B(l,n), l.e. {f(a
c = a ~b the set of fixed points of the permutation a-lb is exactly of the meet a A b. To ,Tl, .•• ,T r- 1,T r
c,
the domain
So the semi lattice T(G) is partitioned by the classes
= G where
for 0';;; i .;;; r-l,
M. Deza
304
T.1
{a
1\
b: a, bEG, f(a
-1
b)
= t .1} = {c E T(G): lel=
L}. 1
r
DEFINITION: Let G C Sn and G be a B(L,n); let T(G) = i~oTi be the above-described partition of the meet-semi lattice T(G) generated by G.
Then we call T(G) permuta-
~'ion geometry if for any a E Ti (0 " i < r) and for any point b(i .e.
where a and b are disjoint (i .e. a 1\ b
=
Ibl
=
1)
and a v b defined) there exists a
unique c E T. 1 such that a vb" c. 1+
The above definition gives a permutation (more exactly, bijection) analog of perfect matroid-design.
For example, any sharply t-transitive set G, G c S , r n generates a semilattice T(G) = .u T. where T. = {c E Si : lei = i} for all 1 =0 1
o~
n
1
t-1, T = G. So, this T(G) is a Perm G(L,n) with L = {0,1, ••• ,t-1}; the t blocks of S(t,k,v) work as hyperplanes in the lattice of matroids in a similar
way.
i
~
The definition of Perm G(L,n) given in [31 was more general; but (because of
Proposition 3.4 in 131 ) the above definition (which is much simpler) excludes only the trivial permutation geometry T(G
= Sn ).
In [31, [101 bijection (and transformation) analogs of general matroids were defined and given their first elementary properties; we will develop this subject in another paper. It is easy to check the following elementary properties of a permutation r
geometry T(G)
T. which is Perm G(L,n):
.U
1=0 1
1 (from now we will suppose, withiut lost of generality, that o T = {}, i.e. E L); o 2) if a,b E T(G) and a v b defined, then a v bE T(G); 1)
T
°
3) any a E T(G) is the least upper bound (in T(G» 4) for any a
E
G define f~a
=
{ii:
"b "a, bE T(G)} , then Ma is the
lattice of flats of perfect matroid-desiqn; 2
5)
IT.1 I
=
i =1
II j=o
(n- ~. ) _--><J_ if 0 " i " r- 1 , c - ~. 1 J
The equalities 5) (which follow from 4»
IGI
IT r I
show that any subset G c S
n
generating a permutation geometry is a B(L,n) of the size same as the order of
(L,n)-sharp group.
The first cases of Perm G(L,n) to study are L
~IIEL(n-~.),
i
{O,.q and L
i.e. the
1
{O,l,t}; i.e.
New generalizations of sharply t-transitive groups
305
an analog od the matroid of rank 4, i.e. of planar space. From now on we consider in this section only special case of geometric group G, i.e. a subgroup G of S , such that T(G) is a permutation geometry, say n
Perm G(L,n).
Of course, any geometric group generating Perm G(L,n) is an (L,n)-
sharp group (see
3).
§
(An example of ({O, q}, q2)-sharp but not gemetric group
is the group {(x,y)
--+
(x+c, ax + by + d)}
of permutations of 2-dimensional vector space over GF(q) with a,b,c,d b
E
GF(q),
"i 0).
For geometric group G, all matroids M , a E G (defined in 4) of the above a proposition) are equal. We denote this matroid by M(G); matroid M(G) is geometric iff 0,1
L, i.e. iff G is 2-transitive group.
E
the fixed-point sets of subgroups of G.
The flats of M(G) are exactly
G is a subgroup of Aut M(G).
Further, G is a Jordan group, i.e. a transitive group with a subgroup fixing some points and
tra~sitive
on remaining points; moreover, the pointwise
stabilizer of any hyperplane of M(G) is sharply transitive on its complement.
In
fact, any Jordan group acts as a group of automorphism of some matroid such that the pointwise stabilizer of any flat is transitive on its complement. Using the above observations one can reformulate a theorem of Kantor ([ 15) ) as the following classification of all 2-transitive geometric groups (Le. with 0,1
E
L); 2r -1, L = { 0, 1,3 ,7 , ••• ,2 r-l - l} ; AGL (r-l,q), n = qr-l , L = {O,l,q,q 2, ••• ,q r-2 };
1) G = GL ( r , 2), n 2) G = 3) G 4) G
=
15, L = {0,1,3}; G = V16 .A , n = 16, L = {0,1,2,4}; 7 .harply t-transitive, La {O,l, ••• ,r-l}. , n
=
ere is a striking similarity of above list to the list of all known perf
matroid-designs (with 0,1
L):
E
1) truncations of projective geometry PG(d,q), d-i -1 1 L = {O " 1 q+l , • • • q q-l -} ' k
=
qd-i -1 d+ 1 1 v = -q--- . q-l' q-l'
2) truncations of affine geometry AG(d,q), L = {O,l ,q,q 2, ... ,q d-i-2 }, k
=
qd-i-l
v
=
qd
3) finite commutative Moufang loops of exponent 3,
306
M.Deza
m L = {O) 1)3 }, k = 9, v = 3 ; 4) Steiner systems S(t,k,v), L = {O, 1 , ••• , r- 1 }.
So, for example, in the first interesting case, L = {O, 1,3} (and so both n, k are
= 1,3
(mod 6)) a geometric group of the above list exists iff n = 7,9,15
(cases 1), 2), 3)) while a perfect matroid-design exists for k = 7,9 and the first unknown cases are k = 13 (v = 133, 183, ... ), k = 15 (v = 171,183, ... ,), k = 19 (V = 307, ... ).
The problem of classification of 1- but not 2-transitive geometric group 0.e.
oE
L, 1 ¢ L)) is open; for permutation geometry,we do not have the same
procedure of geometrization as for matroids.
Concerning the simplest such case of
geometric group of type L = {O,t}, we know only that it is exactly a group of order n (n-t) having TI-subgroup H (i.e. for any g
E
G, either Hg=H or H II H =.+l)-wise generalization of Theorem 2.1 is unknown.
Finally, let us come back to the pennutation geometry T(G) generated by the geometric group G.
Then(T(G),o) is an inverse semigroup whose semi lattice of
idempotents is (semi1attice-) isomorphic to the lattice of flats of the matroid M(G).
Moreover, the semi group (T(G),o) is fundamental (i.e. equality is the only
idempotent-separating congruence) iff 0,1
E
L. The structure of T(G) as a
permutation geometry can be recovered from its structure as inverse semi group and also, it is possible to decide, for a given inverse semi group, whether it arises from a geometric group in this way.
307
New generalizations of sharply t-transitive groups
5. DIGRAPHS ADMITTING SETG (or SET) As was defined at the end of
§
2 a digraph r
= (V,R) admits a SETG (or SET)
G if G is a subgroup (or subset) of (transitive) Aut r which is regular (i.e. sharply l-transitive) on R.
Being both vertex- and edge-transitive, the digraph r
is: a) regular (i.e. the indegree and outdegree of any vertex are equal to the same number d(r), called the degree of r; b) IRI = Ivl. d(r); c) r is either an undirected or an oriented (partial tournament) graph; d) in the case d(r) > Ivl / 2, the digraph r is undirected and, moreover, forodd IVI the number d(r) must be even. An example of digraph r with SETG is a l-regular graph (in terms of [18], i. e. G = Aut r, and, in particular, an elementary circuit of degree 1 and an elementary cycle of degree 2.
(In fact, we will classify below all digraphs of
degree more than Ivl -3 with SETG). The direct product r of digraph r and r 2 admitting a SETG (corresp. SET) l admits a SETG (corresp. SET) and d(r) = d(r l ) d(r 2).
= (V,R) be a digraph admitting a SETG G. Then:
Let r
a) if d(r)
~
Ivl / 2 then r is undirected (i.e. equal to its converse);
b) conjecture; if d(r)
~
Ivl / 2 and G is primitive then either r
=k \
lV (and G is sharply 2-transitive) or G is A5 acting on unordered pairs for L(k 5), i. e. the complement of Petersen's graph (IVI g 10, d(r) = 6); c) if d(r) > Ivl - 1 - ~~ then r is a compete multipartite graph
k~, Ivl = R.m; d) the digraph r -
(~n-l)
2
=n -
-
2~n +
= (V,R) = k~11 x k~11 has (if IVI = n = p2a) d(r)
g
1; it is the largest known degree of digraph which is not
a kR. but which admits a SETG. m
We define d(n) (corresp. d'(n)) admitting a SETG (corresp. SET)}. 2 ~ d(n)
~
d'(n)
~
=
max {d(r): r is digraph (V,R), IVI
=
n,
Because of Proposition 2.5 we have
n-l with d'(n) = n-l if and only if there exists a sharply 2-
transitive subset of Sn (and hence a projective plane PG(2,n)) and with d(n)
=
d'(n)
=
n-l if and only if there exists a sharply 2-transitive subgroup of
S (and hence a projective plane PG(2,n) over a near field which is possible if n
and only if n
= pa,.
308
ftI. L>eza
The known lower bounds on den) are: a) d{n 1nZ) ? d(n l ) d{n Z) (it comes from the direct product fl x fZ; compare with r~cNeish's lower bound min (N(n l ), N(n ) for the maximal number N(n) of Z pairwise orthogonal Latin squares); S ai S a. b) den = i g1 Pi )? i~l(Pi 1 - 1) (it comes from a)); it becames equality a for all n = p and as the smallest other n we have d(Z8) = 18) and then den) > Cn/log log n for any n and some constant C; d) conjecture: den) ? Cn for any n and some constant C; d) if conjecture c) is true then C ~ l/Z since d(Zp) = P for any prime p
> 7;
2 e) den) ? n/2 if n is even (because any km admits a SETG) and den) ? Zn/3 3 if n i5 divisible by 3 because any km admits a SETG); f) den) > n/2 for all n < 5.7.11.17.19.23.29 = 82944785; d(llO) > 72 (from the Mathieu group Mll acting on unordered pairs); d(q(q+l)(q2+q+q+l )
?
q2(q_1)2
for prime powers q (from PGL(3,q) acting on ordered pairs of points of the projective plane). The known upper bounds and exact values of den) are:
= n - 1 iff n = pa; den) 2a a for any n = (2 _l)(2 +l) with odd a; a) den)
~ n - Vi1 if n # pa but den) > n - ~i1
= n-2 iff n = 6 or 14 (and d'(6) = 4, d'(14) = 12); c) den) = n-3 iff n = 15 or 24 (b) and c) come from k~ and k~ admitting b) den)
geometriC ({O,2},2 t )- and ({O,3},3t)-sharp groups as SETG; to compare b) and c) with similar "extention" results in nets and orthogonal Latin squares); d) d(2p) = p for any prime p > 7; d(63)
= 56; d(78) = 72, d(455) = 448 etc.
Table 2 The values of den) for n < 30, n # pa • n
6
10
den)
4
6
digraph realizing den)
a
12 14 15 18 20
21
22
24 26
28
8 12 12 12 16 14
11
21
18
2 k3 L(k ) k3 k7 k5 k3 k5 k2 kll 4 2 3 6 4 7 2 5
13
8 k k2 k4 x k7 3 13
309
New generalizations of sharply t-transitive groups
(The examples of SETG G in above table are the group of rotations of the octahedron for k~, the alternating group A5 for the line graph L(k 5) of k5 (the complement of the Petersen's graph) etc.). 1 From now on we consider complete multipartite digraphs k admitting SETG. m
Some examples are: a) ~ = 2,m-arbitrary; b) ~ an odd prime, m = S(~-1)/2 for some S> 1 (in particular, ~
= 3, m
arbitrary) ; c) ~ = 4, m = S2 where S is a product of prime powers congruent to 1 (mod 3); d)
~
=ma
e)
~
= q+1, m = q-l where q is a prime power;
f)
~
=
g)
~
h)
~
prime power;
q+l, m = (q-1)/2 where q is a prime power with q
4);
q + q + 1, m = q(q-l) where q is a prime power; = 22a + 1, m = 2a - 1 for a odd. =
For all (except the case q> 3 of g)) pairs SETG G.
=3 (mod
2
(~,m)
above we have a geometric
For the case a) we have also a nongeometric SETG G.
~
on k ) has IGI = n(n-m), n = m
~m.
Any SETG G (acting
If, in addition, G has type {O,m} then G is
{O,m}-sharp and, moreover, geometric (see
§
4).
G is geometric for all prime m which are less or equal to the smallest prime divisor of
~-l.
But for any geometric ({O,m}, n
= ~m)-sharp group G the
digraph (V,R) with V = {1,2, ..• ,n}, R = {(a,b): Ga does not fix by b} is k~ and G acts on (V,R) as a SETG. The problem is to classify all pairs (~,m) for which k~ admits a SETG. If m 2 3 m is fixed then we know that k , k always admit a SETG but we conjecture that m m for ~ > 3 the set {m: km admits a SETG} has density zero.
For the other case of fixed 1 the basic result was ([ 17)) that k~ admits a
SETG iff
~
= 2,3,4 or 7. Below we give (from (1) ) two generalizations of it.
THEOREr~ 5.1: For any m;;' 2 the set {~: k~ admits a SETG} is finite. m
THEOR£I.1 5.2: For any prime m;;' 2. the digraph k~ admits a SETG iff R, 1:11 2,3, m+2 m 2 a (with m = 2 - 1), 2m + 2 (with odd m and 2m+1 = pal, m + 2m + 2 (with m+1 an odd power of 2), mt (with t exeept
R,
=
mt for t > 1
and
~ m) or ~ = 7, m = 2 (examples are known for all eases m odd).
310
M. Deza
T. Ito (personnal communication) classified all groups G acting as SETG on £
k and {(O,m}, n = £m}-sharp (we remind,that such (O,m}-sharp groups are, morern over, geometric) for all p~ime m> 2. (For m = 2 it is just the list of groups given by Tsuzuku in [17J ). 1) (Zm x Zrn)
l'l
There are exactly five groups:
Z2 (£ = 2, any m);
2) (Zm x Zm) '" S3 (£ = 3, any m); 3) (Zm x Zm)
1 0 )0
{(":
U = m), the right factor is a subgroup of GL(2,m)
,.,)}
which acts naturally on Z x Z considered as vector space of dimension 2 over m m this field; a
4) Zm x PSL(2,q) (t = q+l, m = (q-l)/g.c.d.(2,q-l), q is a power p of a prime pl. 2 2a+l 5)Zm xS(q)(£=q +l,m=q-l,q=2 ). z For all parameters l,m (except pending case £ = mt, t> 1, m> 2) there are a group of above list but there are also non-{O,mrsharp SETG G on k£. m
Table 3 k£ admitting SETG with t,m ~ 10 m
the sign :', denotes that d(k£) = Rm-m=d(£m)). m
"'-,
~9.i
m~1 2 3
4
2
3
I +
+
I+ I I I
+
6
I+ I+
7
,,, +
5
I I
I
4 :',
+
7
9
10
+
:'~
'O',
+ +
8
'O':
+
+ ..+:
6
5
+
+
?
?
?
?
?
:',
+
+
+
+ ~';
+
?
?
?
?
+
~':
+
+
+
!
8 9 10
I+ +
I
!
i+
+
?
?
?
+
+
?
+
?
+
?
+
?
?
+
?
+
?
?
?
?
+
?
?
:':
I+ I
+
We remark now that the problem of recognizing digraphs admitting a SETG can be formulated in the language of group theory.
Let G be a group with proper sub-
311
New generalizations of sharply t-transitive groups
-1
group H and element g E G such that H n g Hg
= . Define digraph (V,R g ) where V is the set of all left cosets of H in G and Rg = {(xH, xhgH): hE H, xH E V}, then digraph (V,R g ) admits G as a SETG (with G acting on V by left multiplication) and the subgroup H of G is the stabilizer of the vertex HE V. Any digraph admitting SETG is isomorphic to the digraph constructed as above.
As corollaries
we have: a) d(n) = max{IHI: H is a subgroup of index n of G and H intersects some G-conjugate of itself trivially}; b) the digraph admitting a SETG G is k~ if and only if for any x E G there -1
m
exist hl , h2 E H such that x = g hl gh 2 or x = hl g h2. Described above the sugroup H (of the group G) having H n g H g-l = (i. e. hlg
~
g h2 for any hl f h2 E H) for some g E G, can be seen as a permutation
group-code correcting the error consisting of possible multiplication of its elements (from the left or from the right) by the given element g E G.
If g is a
cycle we can see H as a synchronization code. Finally, a generalization of d(n) is dt(n) = max d(r) where r = (V,R) is a uniform directed hypergraph on IVI = n vertices such that Aut r contains a subgroup G which is regular on R.
Then dt(n)
~
n-l with equality iff there exists a
sharply t-transitive subgroup of Sn'
6. DIVERS RELATED PROBLEMS A) One could try to approximate sharply 2-transitive groups by the numbers max IGl/n where H is a Frobenius subgroup of Sn, i.e. a B({O,l},n)-subgroup. But a· S a. it is easy to see that max IGr/n = g.c.d. (Pi 1 - 1) where n = i~l Pi 1 is the prime factorization of n.
So max IGl/n ~ min (p~i -1), but the well-known lower i
1
bound for the largest number of pairwise orthogonal Latin squares is exactly McNeish's bound min(p~i -1). i
1
B) One could try to approximate sharply 2-transitive subsets of S for small n
n f pa (i.e. 6,10,12 .... ) by the numbers IP(n)l/n (where P(n) is a largest 2packing. i.e. a B( {O.l }.n)) and IC(n) lin (where C(n) is a smallest 2-covering containg i.e. a 2-transitive subset of S). n n = 6 IP(6)1/6 = 3 ( d2 > ..... > dp' kl k2 k The notation Gn dl d2 ••••.. dpP will be used throughout this paper to specify briefly the vertex degree sequence of a graph Gn with the above mentioned
=
properties. Let Fn, k be the classes of graphs GR following conditions:
=dkll
k
d P which satisfy the p
(i) G are maximal planar, n
(ii) G have k vertices of degree 5 and n-k vertices of degree more than 5. n
In [7J Schmeichel and Hakimi proved an existence theorem related to the kl k2 k graphs Gn E Fn, k such that : Gn 7 6 5. This theorem, which was extended by the author in [lJ, reads as follows:
=
344
S. Fanelli
THEOREr~
1.1
kl
G n
=7
(see [7) Theorem 2, Theorem 3; (1) Theorem 3): The classes
1 for a E m, bErn'.
B1r(m,i) and BIr(m',j) are disjoint blocks belonging to H(i) and
H(j), respectively.
If P is a plane containing BII(m,i) and Brr(m' ,j) in 0, a
block of type r, Br(x), must be a line of P for any point x E F.
From the rela-
tion between an affine plane and a transversal design TD(q,q), we can say that there exist a plane in 0 containing BrI(m,i) and BII(m' ,j) if and only if T = {Br(x); x E F} forms a transversal design with groups BIr(mi,i), i
= a,l, •.. ,
q-l where mo ,ml, .•• ,mq- 1 are lines of G including m and m'. Suppose that m is parallel to m' in G but m F m'. There is a plane P' in G which contains m and m'.
Let S be a parallel class of P' containing m and m'.
{c(x); x E F) is the set of lines of P' excluding S. i E U.
Let Si={EBr(x); xEF},
If Si has two elements <x,i> and where x and yare in distinct lines
of S, then line xy in G meets m and m', that is, xy is a c(a) for some a E F. a block containing <X,i> and must be of type Ir. block of type rr.
Si = BIr(mi,i), i E U.
with groups B1I(mi,i), mi E S. BrI(m,i), i = a, ... ,q-l. of G such that £ and
£'
So any Si' for i E U, is a
Therefore, T is a transversal design
When m = m', T is a transversal design with groups
Suppose that m and m' are skew lines. 9.'
But
There are no two distinct lines £ and
both meet m and m', and £ and £' have a common point
not on m nor m'.
Therefore, there are no two distinct block of type I, Br(x l ), Br (x 2) such that Br(x l ) and Br (x 2 ) have a common point in H(k), k F i,j, and Br(x l ) and Br (x 2) both intersects BrI(m,i) and Brr(m',j). So, there is no affine subdesign in 0 containing BI1(m,i) and B11(m' ,j). Suppose m and m' have a point c in common.
Let P be a plane in 0 containing
Brr(m,i) and BrI(m' ,j), i # j. Let xa and Yb be points in P-£", at which meets , a E m-{cl, and meets , bE m'-{c}, respectively. For any m-{c} and bE m'-{c}, c(x a ) = m and c(Yb) = m', respectively, since ca = m and cb = m', respectively. Let Ak = {z, E BI(x a ), a E m-{c}} and Bk =
a
E
E Br(Yb)' bE m'-{cll, where k # i,j.
iZ;
Yb
~
Si nce xa # xa , if a -; a', and Yb, if b F b'. then Ak = m-{c} and Bk = m'-{c}. Let d be a point in P-£", at
Affine geometries obtained from projective planes
which meets , a
<e,k>
E
m-{c}, b E m'-{c}.
E
Bl(d), is not in mUm'.
ments in H(k), k BII(m',j).
F i,j.
Then point e of G, where
Therefore, P has at least 2(q - 1) + 2> q ele-
So there is no plane of 0 containing BII(m,i) and
D
LEMMA 3.2: Suppose 0 has the ppopepty (1) fop c(x). vapiety e
361
F B of
Fop any block B and any
0, thepe exist a unique (q2,q-l)-subdesign (plane) of 0 contain-
ing B and e.
PROOF: First, we show that if a (q2,q-l)-subdesign (plane) S in 0 contains a type I block B = {; i 1
H(i), i
E
U.
E
U}, then S contains exactly one type II block from each
There is a variety in S but not on B, say e
II block C = BII(bka,k) must be a block in S. containing e and disjoint to B.
= . Then, a type
There is a unique block B' in S
If B' is a block of type II, which is a block in
H(k), then two distinct blocks B', C having a common variety e are contained in S. B' and C generate a plane in H(k), which implies that S is in H(k).
But S has
points not in H(k). Therefore B' must be a block of type I, say B'
= {; 1
i
E
U}.
Then S has at least one type II block BII(bibi,i) in each H(i), i
By the same manner above, S cannot have two type II blocks from a H(i).
E
U.
Further,
Lemma 3.1 says that there is a plane containing BII(£,i) and BII (£' ,j) i F j if and only if £ is parallel to £' in G. Then, blocks of type II BII(£i,i), i E U contained in S have the property that the £.'s are parallel in G. 1 Suppose B is a block of type I {; i 1
j
E
U, there is unique line
£~
J
E
Uland e
=
then, for any
containing b. and parallel to Bka in G.
From the
J
proof of Lemma 3.1, a plane can be constructed from those parallel lines. Suppose B = BII(£,i) and e
= ,
i
F j. If there is a plane containing
Band e, it must contain a block of type I, say C, such that e, is a point of £. £.
E
C, where b
There is a unique line £' in G which contains a and parallel to
By Lemma 3.1, there is a plane containing Band BII(£',j), which is the unique
plane by the structure of S mantioned above. Suppose B = BII(£,i) and e = , then there is a unique plane in H(i) containing Band e since H(i) is isomorphic to AG(n,q). D Since G is an affine geometry, G has the parallelism on lines satisfying the axi oms of an affi ne geometry.
We denote para 11 eli sm in G by I IG.
a relation liD on blocks of 0 using II G•
Here, we define
R. Fu;i-Hara and S.A. Vanstone
362
For any blocks B,B' of 0, BIIOB' if and only if (a) Band B' are both type II blocks, say B = BII(i,i), B = BI I ( 9' . , E U, S uc h th a till G £ ' • . ,1") , 1.,1
or (b)
blocks, say B = {; i U}, satisfying:
Band B' are both type B'
= { 2.
We use a method in Hughes and Piper [9] for coordinatization.
The points in p-~oo are assigned coordinate from {(x,y); x,y on
~oo
365
E
GF(qn)}.
The points
are coordinatized as
= {( oo) } u
~ oo
{( a);
a
E
n
GF (q )}
such that line (O,a)(l.O) meets (a) and the line (0,0)(0,1) meets (oo). The points in p-~oo can be thought of as elements of GF(q2n) provided a suitable primitive irreducible polynomial over GF(qn) is selected. . element of GF *(q 2n ) = GF(q 2n )-{O}. Then. tlve GF"'(q2n) Let
~o
{Xi; i
=
=
Let x be a primi-
0.1, .... q2n_ 2 }
be
~
o
{oo} u {X j {(O,a); a
where'"
=
(0.0).
; j EI E
O(mod qn+l)}
GF(qn)}.
£0 is incident with £oo at (oo).
We define an affine geometry G
of dimension n and order q by: (i) a point of G is a point of £0' (ii) a line xy of G passing through x and y. X,y
E
£0' is defined by
xy = {Ax + (l-A)Y; AE GF(q) • Consider two types of collineations a,S on n+ ) a: x ... X(q l • X E 'P-l oo •
p-~oo
defined by
where a E ~ • o n C = {a o, a l ••••• a q -l} and C2 = {S ; a E £ } form collineation groups on P-£ . l a 0 "" Any j E C fixes lines of P-£"" passing through oo, and any Sa E C2 maps a line £ l passing through", to its coset. Therefore. any collineation of these two types fixes £oo pointwise, and fixes £. These collineations also act on G. S. 0 a for any a E £o • fixes the points on ~ oo pointwise. Let a point x E P-£ oo be on a line £ of P-~ incident to point z = £ n £ • then {S (x); a E £ } is the set of points on "" 0 oo a 0 £.
Let point x E P-£oo be on a line m of P-£oo passing through oo, then
{ai(x); i = 0,1, .... qn_ 2} is the set of points of m-{oo}. point x E P-(£ oo {a
i
Sa(x);
U
a
i
£0 ), Sa
E
Cl
x
C2 }
So, for an arbitrary
366
R. Fuji-Hara and S.A. Vanstone
is the set of points of P-(£ u '"
t
)_
0
Let BI(x) = (, •••• ,J be a block of type I of D. a(BI(x)) is also a block of type I of D; '"I(BI(x)) Let BlIP,.i)
BI(cl(x))
=
Then,
{., ••• ,J. 1
(,, •••• J be a block of type II of D, then l a(Brr(t.i))iS also a block of type II. =
,:.(BII(t.i))
BII (a{9,),i) {., ••• ,J.
Similarly, any co1lineation Sa of C2 also acts on D. state the following results: LEf4MA 3.5: If a co:Zinearion a on P which fixes
£",
Therefore, we can
pointUJise and preserves the
a.ffine geometry G cwfined on £0' then a induces an automorphism of D.
Let x be a point on P-(£", a.1
E
m meets x at u1.•
£0) and let m be a line of G.
U
A line a.x, 1
Let D be a BIBD constructed by the previous method with
00
U = {u o .u 1 '···,u q- 1}'
Then any block of type I, BI(y). where y i
be generated from BI(x) by some a Sa
E
C1
x
E
P-(£",
U £),
can
C2 ,
i
Br(Y) = BI(a ea(x)). This implies. for every point x E P-(£
o
U
£ ), c(x) is a line of G. '"
Therefore, we
can state the following result. THEOREM 3.3: If there is a set of collineation C on P of which any collineation of C sat'!:3fie3 Lemma 3.;;, and for any two points x.y
E
P-(£o
U
tJ
there is an a
E
C
c:uch that y = a(x), then the BIBD D constructed on P is isomorphic to AG(n+l,q), ",hel: n
> 2, q > 2.
4. SKEW RESOLUTIONS OF AG(3,g)
Let D be an (r,l)-design having two resolutions Rand R'. orthogonal to R' provided iR n
SI .;;
1
We say that R is
Affine geometries obtained from projective planes
367
R'. Orthogonal resolutions in block designs have recently been of interest to various authors. (see for example [7). [11)). If we can construct a skew resolution of the lines in AG(3,q) then this resolution and the natural resolution from parallelism provide a for each resolution class R E R and each resolution class S
pair of orthogonal resolutions.
E
In the remainder of this section, we construct a
skew resolution of the lines in AG(3.q). In R. Fuji-Hara and S.A. Vanstone [4). orthogonally resolvable (q2 + q+ 1.1) -designs 0 are constructed on a finite projective plane P of order q2 where P-t is coordinatized by two dimensional vectors over GF(q2).
In the plane. we can
choose q points U on too such that the design 0 is isomorphic to AG(3,q).
But any
two of the resolutions constructed by the method of [4) do not necessarily form a skew resolution.
In this section. we show a construction for a skew resolution of
AG(3,q) obtained from PG(2,q2) in Section 3. LEMMA 4.1: Two blocks of type I, Bl(x l ) and Bl (x 2), of 0 are parallel if and only if line x x is incident at the point Z = too n £0' l 2 PROOF: Let Bl(x l ) }.
=
{ and , and let C be a block containing <X,l> and . l Co ~ BIr(xoo,O) and Cl = BrI(Xoo,l). That is, Co and Cl are parallel from Lemma {<x,D>, <X,l>, ••• ,<X,q-l>} since x is on
~o.
But Cl = Y~l (Co) and Co must be skew from Lemma 4.3, contradiction. Therefore, for any i E U, f(X,i} is a set of q + 1 element in H(i) in which
3.1.
no three elements are collinear.
In a H(i}, the number of blocks which has at
least one element of f(X,i) is (q;l) + q + 1
= (q2
+ 3q + 2}/2.
the number of blocks, q2 + q, in an affine plane H(i) when q ~ 3.
This is less than So, there
exists q - points set E on m where f(E.i) is a line of H(i) and does not contain 0), divisible semiplanes (A' = 0, A = 1) and semibiplanes (A' = 0, A = 2) and, when c=l, symmetric 2-designs.
Generalizing to p-divisible designs well known results by Hughes (see [16] on automorphisms of symmetric 2-designs, we show the existence of certain matrix equations which must be satisfied if a p-divisible design is to possess a standard automorphism group G (i.e. an automorphism group with orbits of length either 1 or iGi).
These equations involve matrices with rational entries, and the Hasse-Min-
kowski theory of rational congruence is applied to give numerical conditions on the parameters v=mc, k, A', A, the order u of G and the numbers N of fixed points and
~
of fixed classes. In section 1 we review the relevant points of the Hasse-Minkowski theory.
In section 2 we prove some general lemmas on matrices of a special type and on their Hasse invariants.
In section 3 we establish a rational congruence for a
general incidence structure and we deduce from it a different proof of a theorem by Hughes ([ 16] , 2.1) for symmetri c 2-des i gns. In section 4 we specialize the above rational congruence to p-divisible designs with a standard automorphism group.
We then apply (section 4 and 5) the
Hasse-Minkowski theory to obtain nonexistence theorems which generalize, simultaneously, the well known Bose and Connor, Hughes, and Bruck-Ryser-Chowla theorems.
378
D. Ghinelli Smit
If the dual structure of a p-divisible design is also a p-divisible design (i.e. the design is both point and block-divisible) we call it simply a divisible design.
We also show, in section 4 and 5, that for divisible designs all the
nonexistence theorems can be improved.
Most of the proofs in section 4 and 5 are
simply sketched, but it appears clearly how crucially they depend upon the lemmas proved in section 2 (see [10], for complete proofs). We point out that all the results in this paper concern standard tactical decompositions of p-divisible designs and we do not need to assume the decomposition to be the orbit decomposition of some automorphism group.
If we make use of
the actual group action, it is possible to get stronger results.
In this case we
need a different approach (due, for synllletric designs, to E. Lander [22] (1) which uses algebraic coding theory and modular representation theory.
We note, howev-
er, that these methods do not seem to be applicable for standard tactical decompositions that do not arise from the orbit decomposition of a group.
1. THE LEGENDRE SYMBOL. THE HILBERT NORM RESIDUE SYMBOL AND THE HASSE INVARIANT In this section we give a description of the relevant points of the Theory of congruence of quadratic forms (over the field of rationals) due to Minkowski [26] and Hasse [ 14]. fa
L
Instead of considering invariants of quadratic forms
a .. x.x., we consider invariants to refer to the corresponding matrices lJ 1 J
((a .. )) (for a more detailed account see Jones [19), chapter 2). lJ
Let p be an odd prime.
The integers a 1 0 (mod p) are divided into two classes called quad1~tic residues and quadratic nonresidues according as x2 a
=
(mod p) does or does not have a solution x (mod pl. The Legendre symboZ is defined by (1)
If a
+1
if a is a quadratic residue modulo p;
-1
if a is a quadratic nonresidue modulo p.
(!) p
=0
(mod p) we may write (!) P
= O.
The following theorems give some of the main properties of quadratic resi-
-----
(1) Added in proof: see also E.F. Assmus, Jr. and D.P. Maher, Nonexistence proofs for projective designs, Amer. Math. Monthly, 85 (1978), 110-112, and C. Hering, On codes and projective designs, Kyoto University Mathematics Research Institute Seminar Notes 344 (1979). 26-60.
379
On divisible designs
dues and nonresidues (for proofs see Hardy and Wright [13], chapter VI; Theorem (1.2) is due to Gauss and (6) and (7) are known as the taw of quadPatia reaiproaity) •
THEOREt~
(2) (3) (4)
1.1: If p is an odd prime, then b (mod p) implies (~) p a(P-l)/2=,(~) (mod p) ,
a
='
(E.) p
(~) (E.) p p
(ab) p
THEOREM 1.2: If p and q are two odd primes, then (5)
(.::!. ) p
(6)
(.?)
(7)
(E.) (51.) = (-1)[ (p-l)/2][ (q-l)/2]. q p
p
A slight generalization of the Legendre symbol is the Hilbert p-norm residue symbol (a,b)p (Hilbert symbol, for short).
Where no ambiguity is possible, we
shall omit the subscript p on the Hilbert symbol.
For any two nonzero integers a,
b, the Hilbert symbol is defined to be 1 or -1 according as the (8)
a x2 + b
i
='
z2
congrue~ce
(mod ph)
does or does not have solutions in integers x,y,z not all multiples of p, for arbitrary h. (9)
We also include p = ~ to mean that 2
2
ax+by=z
2
does or does not have solutions in real numbers x,y,z,not all zero. (10)
(a,b)~
Thus
= + 1, unless both a and b are negative.
THEOREM 1.3: The equation ax 2 + by2 zero if and only if (a,b)
p
=+
z2 has solutions in integers x,y,z, not all
1 for all primes p (inaluding p
= ~).
Pall [29] has shown that if a=a'pCl and b=b'pf3, ~/ith a' and b' prime to p, then
380
D. Ghinelli Smit
that is, by (5) (12)
(a,b) p = (_1)cxS(p-l)/2 (r)s (%')cx, if p> 2;
(13 )
(a,b)2
=
If p = 2, then
(-1)[ (a'-1)(b'-1)/4)+I s(a,2- 1)/S)+[cx(b,2_ 1 )/S).
The following theorem gives further computational properties of the Hilbert symbol. THEOREM 1.4: The Hilhert symbol has the following properties, where a,b,c,r,s aPe integers
= 1 the product being over all primes p (including p = 00),
(14)
IT(a,b)
(15 )
(a,b)p = (b,a)p ,
(16 )
(ar ,bs) p = (a,b) p ,
( 17)
(a,-a)p = 1,
(lS)
(a,b)p (a,c)p
( 19)
(a,a)p = (a,-l)p ,
(20)
(ar,br)p = (a,b)p (r,-ab)p)
p
2
2
2
(a ,b)p
1,
= (a,bc)p ,
If P is an odd prime,
= 1 if a
(21)
(a,b) p
(22)
(a , p) = (~ ) if a '1 0 (mod p)
(23)
if a1
p
and b are prime to p (see (11)).
P
= a2 '1
0
(mod pi, then (al,b)p = {a 2,b)p'
If x and yare nonzero rational numbers the Hilbert symbol (x,y) p is also defined and has similar properties; in particular, it has the property (24) where
(x ,y)
2
p
2
= (xp ,yo )
p ~~
i:
p, 0
£
Q = Q - {a}. Therefore, if x=a/b and y=c/d, with a,b,c,d
£
Z =Z-{O},
then (2S)
(x,y) p
= (xb 2,yd 2) = (ab,cd)
and actually we only have to consider Hilbert symbols of integers. Now, let A,B be two rational symmetric and nonsingular matrices of the same
381
On divisible designs
order and let A - B mean that A and B are rationally congruent matrices: that is
U A UT = B
(26)
for some nonsingular matrix U with rational entries (throughout the paper we shall let MT denote the transpose of the matrix M). If x and yare nonzero rational numbers such that x/y is a rational square then we write x .:: y. Let u be the order of A and let D. (i=l, ..• ,u) denote the leading principal 1
minor determinant of order i in A.
For our purposes we can assume that all of the
D. are nonzero(2).
Then for each prime p the Hasse invariant H (A) is defined by u-l P Hp(A) = (-1 ,-Du)p i~l (Di ,-D;+l )p'
1
(27)
The major theorem that we shall use is THEOREM 1.5 (Hasse-Minkowski): The necessary and sufficient conditions for two positive definite, rational and symmetric matrices A and B of the same order to be rationally congruent are that (28)
(3)
IAI.:: IBI
and (29)
H (A)
P
= H (B) P
for all primes p (including p a
oo).
By property (14) of the Hilbert symbol, we are able to exclude p = 2 from consideration since H2(A) = H2(B) is a consequence of Hp(A) = Hp(B) for all odd primes p and p = 00.
The invariant Hp may be expressed in a more symmetrical form as follows:
-----
(2) The matrices we need to consider will be all positive definite. Otherwise Pall's generalization of the Hasse symbol can be used, that is: if Dk=O the symbols (Dk,-Dk+l) and (Dk-l,D k) are interpreted to be (±l,-Dk+l) and (Dk-l,+l) respectively with the ± sign chosen arbitrarily. Actually, lemmas 1 and 2 in Jones [191 show that A must be nonsingular and Dk = 0 implies Dk+l Dk- l ~ O. (3) We denote bYIMlthe determinant of a square matrix M. In what if x and yare positive integers, Ix will denote the identity matrix of x, J(x,y) the (x,y)-matrix all of whose entries are 1, and J x = J(x,x). matrlx M (not necessarily square) we define a(M) to be the total sum of entries.
follows, order For any its
382
D. Ghinelli Smit
u-1 H (A) p
u-1 = (-1,-1)
°
where o
a
II
;=0
(Oi+1,D i ) (Oi+l,-l) ,
1; that is u-l
If Au- 1 denotes the matrix obtained by leaving out the last row and column of A, we get from (3~}, (31 )
Hp (A) = Hp (A u- 1) (IAI,-IA u- 1 1 ).
From this and the properties of the Hilbert symbol we can at once deduce the follow; ng theorems (see Jones [ 19J , and Pa 11 [29J).
THEOREM 1.6: If A ,A , .•• ,A are mtionaZ, nonsinguZar and syrrmetric matrices and m l 2 if A = e.A. = diag(AI, •.• ,A ) , then 11m (32)
Hp(A) = (-1,-1)
m-l
m
m Hp(Ai)} {. ~ (IAil,IAjl)}. i=l l,J=l ;<j
{II
As a particular case we have the following COROLLARY 1.7: The Hasse invariant of A = 1m x B (where x is the KPonecker product of matric:es) is
(_l,_l)m-l {H (B)}m (IBI._1)m(m-l)!2.
(33)
p
THEOREM 1.8: For A = a1 we have m
(34)
H (A) = (-1,-1) (_1,a)m(m+l)/2. p
THEOREM 1.9: If
p
is a nonzero rationaZ number
383
On divisible designs
We shall state another lemma on a property of the Hilbert symbol which we shall use later (see Bose and Connor [3] , lemma III). LEMMA 1.10: For any two nonzero rational numbers x,y and any prime p
(X,y) p = (-xy,x+y) p •
(36)
2. SOME USEFUL LEMt-1AS
In this section we prove some general lemmas about matrices.
The notations
are in footnote (3) (see section 1). LEM~1A
2.1: Let 0 be a nonsingular matrix of order u with entries in a field F, and
let d
E
F.
Then
110 + dJ u I
(1)
=
101(1+do(0
-1
)).
If (l+dO(O-lr f 0, then the matrix Z = O+dJ
is invertible and
u Z-l = 0- 1 _ d (O-lJ 0- 1). l+do(O-l) u
(2)
If the above matrices have rational entries, p is any prime (including p
= 00),
and we can consider the Haase invariants (see section 1. footnote (2)),
then (3 )
H (Z) p
a
H (D) (dIOI, l+do(O-l)) p p
or, equivalently (4)
PROOF: We give the proof for matrices with rational entries since the extension of
(1) and (2) to matrices over any field is obvious. matrix
(5)
Clearly
Z'
We consider the ((m+l)x(m+l))-
384
D. Ghinelli Smit
(6)
Iz'l
and by (31) 1
-dlzl,
=
(4)
Hp (Z')
(7)
=
Hp (Z) (IZ'I,-IZI).
Now we consider the ((m+1)x(m+1))-matrix
Z" =
(8)
\ I
0
l-dJ(l,u)
-dJ(U,l)l -d )
Let B the square matrix of order m+1 with all entries 1 in the diagonal and in the last column, and with entries 0 elsewhere. B Z' BT
(9)
Since
= Z",
we have (see (6) and (7)) (10)
IZ" I
=
IZ'I
=
-diZI ,
and Z' -::- Z", that is, by the Hasse-Minkowski theorem. Hp(Z")
(11 )
= Hp(l') = Hp(Z) (lZ'I. -IZI).
On the other hand (12) and, by (31)1' (13)
Hp (l")
= Hp (0)
(ll"I,-IOI).
Compari ng (10). (11) and (12), (13) we get (1) and, since Iz' 1= IZ" I ( 14)
1 H ( l ) (! Z' I , -I 0 I ( 1+do (0- )) = H ( 0 ) (I l' I ,-I 0 I ) • p
P
By (18)1 we can cancel (Il' I .-IDI) in both sides and from (6) we obtain (4).
From
(1). (18)1' (19)1 we get the equivalence between (3) and (4). The proof of (2) is an exercise. Let a, b be ra ti ona 1 numbers, u a pos iti ve integer and, fo 11 owi ng Hughes! 16] • define an (a,b,u)-matrix to be a square matrix of order u with a+b on its main
(4) Here the subscript denotes that we refer to the formula (31) of section 1. We shall use this notation throughout all the paper, when referringto formulae of different sections.
385
On divisible designs
diagonal and b elsewhere.
In other words, an (a,b,u)-matrix is defined by
A = aI + bJ ,
(15 )
u
u
and we can apply our lemma (2.1) in this particular case -1
-1
-1
D =a I u , a(D )
D 1:1 aI , d=b,
(16)
u
=
ua
-1
•
we thus get COROLLARY 2.2: Let A = aI u+bJ u be an (a,b,u)-matrix, with a (17)
IAI
If a+ub
~
=
a u-
l
Then
(a+ub).
A- l = l I _ a u
-1
O.
0, then A is invertible and
( 18)
(i.e. A
~
is an (a
-1
b
a(a+ub)
,-bl a(a+ub)J
-1
J
u
,u)-matrix).
We aZso note that
(19)
a(A)
=
u(a+ub).
Hence
u(a+ub)
(20)
-1
•
The following lemma is an easy consequence of lemma (1.10) and of the 2 property (a ,b) = 1 of the Hilbert symbol. LEMMA 2.3: Let u, a, b be the nonzero rational numbers. (21)
(-uab,a(a+ub))p
Then for any prime p
= 1.
From the Lemmas (2.1) and (2.3) we can at once deduce the following LEMMA 2.4: Let A = aI u+bJ u be a nonsingular (a,b,u)-matrix (i.e. a a+ub
~
0).
Then for any prime p
(22)
H (A) = (-1,-1) (-l,a)
Furthermore,
(22)
p
is equivalent to
u(u+l)/2
u (ba ,a(a+ub)).
~
0 and
386
(23)
D. Ghinelli Smit
H (A) p
(-1.-1) (_1.a)u(u-l)/2 (-uau-l.a+ub)(u.a).
PROOF: By (1.8), (22) is lel1Jl1a (2.1) in this particular case (see (16)).
From
(21), and (16)1' (18)1' (19)1 it follows that (24)
u (ba ,a(a+ub))
«-uab)(-ua
u-l
).a(a+ub))
(-uab.a(a+ub))(-ua (-l.a)(u.a)(a.a)
u-l
u-l
.a)(-ua
(-ua
u-l
u-l
.a+ub)
.a+ub)
u u-l ( -1 .a) (u.a)( -ua • a+ub) • But ( 25)
~:~ (mo d 2) 2 +u- 2
and. therefore, (22) is equivalent to (23). The formulae (17) and. for odd primes. (23) were proved by Hughes (see [16], lemmas 2.1 and 3.1) in a different way.
Formula (22) seems to us more
interestin~
as it gives at once (see (25)) the proof of the Bruck-Ryser-Chowla theorem. In order to calculate the Hasse invariants of our matrices in section 5. we shan need, essentially. only the previous Lemmas. (25) and the following elementary property of congruences modulo 2: (26)
ab{a+b) : ab(ab+l) 2
-
2
(mod 2).
3. A RATIONAL CONGRUENCE FOR TACTICAL DECOMPOSITIONS.
THE ORBIT DECOMPOSI-
TION Let S=(P.B.I) be a finite incidence structure with v =Ipi points and b = IBI blocks. A tacticaZ decomposition of S is a partition of the points and the blocks of
S into classes Pl •.•.• P ,. and Bl ••••• B ,. respectively such that b v (i) the number of points of P. incident with a block of B. depends only on
J
1
and j and is denoted by (P .• B.). and. dually 1
J
(ii) the number of blocks of B. incident with a point of P. is a constant J
(Bj.P i ) depending only on the choice of classes.
1
On divisible designs
Given a tactical decomposition
~
387
of a structure S we consider the diagonal
matrices
and
Obviously tr D(P)
(3)
1:1
v,
Let X be the (v'
w.1 I of
with
its copies.
tr D(B) x
=
b.
v)-matrix obtained from I v , replacing the i-th column Clearly
T X X = D(p).
(4)
Similarly, if Y denotes the (b' column with (5)
IB.I
b)-matrix obtained from I b, replacing the j-th of its copies, then x
J T Y Y = D(B).
Let A be an incidence matrix for S associated with the given tactical rows of A correspond to the points of Pl , the next iP2i rows to P2' etc. and similarly for the columns). If we denote by L the
decomposition (i.e. the first (v'
IPl I
b')-matrix whose (i,j)-th entry is (P.,B.), we have
x
1
J
X A = L Y,
(6)
and, by transposition of matrices
Hence:
Equation (8) is particularly interesting when L is a non singular square matrix. In this case, which is always valid when A is nonsingular (see Block [1), cor. (2.1)), (8) establishes a rational congruence between the diagonal matrix D(B)
= yyT and the matrix X(AAT)XT. Thus (8) permits the application of the
theory of Hasse-Minkowski outlined in section 1. The most important example of tactical decomposition of a structure S is
388
D. Ghinelli Smit
given by the orbits of any automorphism group G. We say that S is a square nonsingular struature if it has a square nonsingular incidence matrix.
In this case, it follows from a result of Brauer [5]
that any automorphism group of S has equally many point and block orbits, thus v' = b'
(9)
and L is a square matrix (see also Parker
[30] ,
Hughes [ 16]
,
Dembowski
(6),
LUneburg [ 241 ) • We choose in each point-orbit base bloak x. (i,j=l, ••. ,v').
p.
1
J
J
1
IG p . I = r., 1
IG
1
1
x.
1
which fixes P., and, respectively, x .• ( 10)
a base point P. and in each block-orbit a
Let Gp . and respectively G ,be the subgroup of G
I =
Xj
J
If
5.,
J
then each point in P. has a stabilizer of order r. and each block in B. has a 1
stabilizer of order
J
1
5..
J
We consider the diagonal matrices of order v'
(11 )
and we put IGI (12)
=~.
D(P) =
With the notation introduced in (1)-(8), we have ).l
-1 R ,
D(B)
= ).l
-1 S ,
and thus ( 13)
For our purposes it will be better to consider, instead of the matrix l = (( (Pi,B ) )) the generalized inaidenae matrix, that is the (v'
Ha
j ((h ij )) where hij = I{g
E
G I Pig
x
v')-matrix
Xjl!.
Clearly (14)
H=RL
and (8) becomes (15)
H S-l HT
= ).l-1 R X (AAT) XT R.
In spite of its appearance, (15) is, in our case, simpler than (8).
For
example, when the structure is a symmetric 2-design 2-(v,k,A) of order n = k-A, then (16 )
A AT
=n
I + AJ v
v
389
On divisible designs
Hence
But it is easy to see that (18)
so that, from (13), we get T
(19)
T
-1
-1
X (AA ) X =)J R (nR+)JAJ v ') R
and (15) gives
-1 HT = nR + )JAJ ,.
HS
(20)
v
This is a different and very easy proof of theorem 2.1 of Hughes [ 16).
Equation
(20) establishes a rational congruence between the diagonal matrix S-l and the matrix nR + )JAJ v , (which is of the type considered in Lemma (2.1». Hughes in [16) applies the Hasse-Minkowski theorem to (20) under the hypothesis that G is a standard automorphism group.
We note that, using lemma (2.1), it is not difficult
to write the Hasse-Minkowski conditions for more general
automorphism groups (see
(10) and compare with Dembowski [8) , theorem (2.1.11), [6) and [7), Lenz (23), p. 117, Block [lJ, theorem 6.1, Goldhaber [111, p. 391). An interesting generalization of symmetric 2-designs are the structures introduced by Bose and Connor [31, as Symmetric Regular Group Divisible designs. The main result in [31, consists in considering the rational congruence given by (8) when the tactical decomposition is trivial.
In the next sections we consider
those structures and apply the Hasse-Minkowski theory to (15), to give numerical conditions which must be satisfied if such a design is to possess a standard automorphism group.
4. RATIONAL CONGRUENCES FOR P-DIVISIBLE DESIGNS.
FIRST NONEXISTENCE
CONDITIONS DEFINITION 4.1: A point-divisibZe design '(p-divisible for short) is a tactical configuration for which the v points are split into m classes of size c, such that points have joining classes.
nu~ber
A' (resp.: A) if they are in the same (resp.:
differen~
390
D. Ghinelli Smit
The study of these designs commenced with the work of Bose and Connor [3) who introduced them as an important subclass
of the Partially Balanced Incom-
plete Block Designs with two associate classes of Bose and Nair [4).
Bose and
Connor use the term "group" instead of "classes" and the term "group divisible design" (shortly GD-design).
We avoid this terminology as there is no connection
with the algebraic group notion and as we will deal with automorphism groups of the designs. Since the appearance of (3), much research has been done in connection with point-divisible designs.
For a survey of known results see Raghavarao [32),
Dembowski [8) (pp. 286-289, where the designs are called "divisible partial designs"), Jungnickel [20) and C. Mitchell [27). For sake of brevity, we will assume, in this paper, the following: DEFINITION 4.2: A p-divisible design D=D(m,c,k,A',A) is an incidence structure which satisfies (4.1) and is: (i) square (i.e. with b=v=mc blocks), (ii)
(i.e. it has a nonsingular incidence matrix, or, what is the same, k f A' and k2 f vA).
nonsingu~al'
When the dual structure
D'·:
is also a p-divisible design (i .e.
D
is both
pOint and block-divisible) we will say that D is a divisible design. Necessary and sufficient conditions for a p-divisible design to be divisible are given by C. Mitchell in [28); in particular he shows that if
D
is divisible,
then D* is a p-divisible design with the same parameters. We note that in Bose and Connor's terminology our "p-divisible designs" are called "Symmetric Regular Group Divisible Designs" (SRGDD, for short), and, by the above mentioned result by Mitchell, our "divisible designs" are precisely the structures that Bose in [21 calls "SRGDD with the dual property". For divisible designs we shall use the following result, stated by Bose ([ 21) in equi va 1ent form THEOREM 4.1: Let D = D(m,c,k,A' ,A) be a point classes
p] •..•• ?m
divisib~e
design.
and the divisibility block cZasses
tacticaZ decomposition of D and the matrix
Then the divisibiZity
B1••••• Bm form
a
391
On divisible designs
(1 )
« (P. ,B.) »
L
J.
1
satisifies (2)
L LT = n1
(3 )
L J m = J m L = kJ m,
m
+
CAJ , m
where
(4)
n
= (k~A')
+ C(AI -A).
Note that (2) establishes a rational congruence between the (n,cA,m)-matrix LLT and the identity matrix 1m' From our (2.2), (2.4), we get easily, applying the Hasse-Minkowski theorem, the following THEOREM 4.2: If a divisible design D=D(m,c,k,A',A) exists, then (i) if m is even. n is a square; (ii) if m is odd, the Diophantine equation
n x2 + (1)(m-l)/2 c", 2 y
(5)
= Z
2
has a solution in integers x,y,z not all zero.
If m and c are both odd, Theorem (4.2) appears to be an improvement of Bose and Connor [3J, theorem 9. REMARK 4.3: An interesting special class of divisible designs is that with AI
= 0,
A> O.
In this case p-divisible is equivalent to divisible.
These de-
signs are also called Divisible Semisymmetria designs or, when A = 1, Divisible semiplanes, when A = 2, Divisible Semibiplanes, (see [9] and [10], or Hughes [18),
Jungnickel [20J, Wild [33) ).
SemisYlTllletric designs with c
=
1 are symmetric 2-
designs. For semisYlTllletric designs L is a (O,l)-matrix and (2)-(4) show that L is an incidence matrix for a symmetric 2-(m,k,cA). Then, it is natural to call this 2design the quotient design(with respect to the divisibility relation).
In this
case, theorem (4.2) is the Bruck-Ryser-Chowla theorem for the quotient design. Mielants gives in [25) a direct construction of the dual of the complement of the quotient design 2-(m,k,cA), and applies the Bruck-Ryser-Chowla theorem to
392
D. Ghinelli Smit
get nonexistence conditions. the particular case
~'
= 0
Therefore the conditions in (25) are equivalent to
of (4.2) (see, also, (21) and Payne [31), theorem
(5.1) ).
REMARK 4.4: In general l is not a (O,l)-matrix but, by abuse of terminology we will consider l as a sort of pseudoinoidenoe matrix and, also in general, we shall speak of the quotient symmetrio pseudodesign which is a 2-(m,kH'(c-l),o).
(6)
Going back to p-divisible designs (not necessarily divisible), let
Abe
an
incidence matrix for D=D(m,c,k,A' ,A) associated with the division (i.e. the first crows of
Acorrespond
to the points of the first class
poi nts of the second class
PZ'
etc.).
(7)
A AT
(8)
A Jv = Jv A = k J v
where
x
= I
m
x
C + AJ
Pl ,
the next c rows to the
Then
v
is the Kronecker product of matrices and C is the (k-A',A'-A,c)-matrix
(9)
C = (k-A')l c
+
(X'-X) J c •
We will call order of the design the integer (10)
n = (k-t.') + CP'-A).
It is easy to prove (see (3.1) in (3) (11 )
that
2 n = k - >v.
Now, let G be an automorphism group of D, and let A be an incidence matrix associated with the orbit tactical decomposition.
With the notation of section 3,
we have
Since A and
Aare
both incidence matrices for D, there exist permutation matrices
P and Q such that (13)
A = P AQ.
Hence, from (7), we get
393
On divisible designs
( 14)
and, for our purposes, the nature of the permutation Q on the blocks is irrelevant. As in section 3, we have
X J XT = ~2 R- l v
R- l ,
J '
v
and, therefore -1
T
H =~
HS
(15)
-1
(RX) P (1m
T
C) P (RX)
x
T
+ ~A J
v'
From now on G will be a standard automorphism group. points fixed by G.
Since D is nonsingular, v'
Let N be the number of
= b' which implies that N is also
the number of fixed blocks. Let
be the number of fixed classes, and let i be an integer, 0
~
~
i
~
c.
We define
={(m-~)/~}H
v'={(v-N)/~}+N
is the num-
is the number of orbits on
cZasses.
REMARK 4.7: As it should be, when
~=l
(~v'=v=mc,
u=m)
(28) gives, as a particu-
lar case, that (k_A,)m(c-l)(k 2_VA)m-l must be a square,
(29)
that is theorem 7 of Bose and Connor [3]. If c=l, that is the design is a symmetric 2-(v,k,A), then v=m,
~
=N, and
thus v'=u and (28) asserts that v'-l n is a square
(30)
that is the condition of Hughes [ 16] • If
~=c=l
we have, of course, the first condition of the Bruck-Ryser-Chowla
theorem (see
~1.
Ha 11 [12] ):
(31)
v-l n is a square.
I
We can now use condition (29) to simplify, when m is even or m is odd but c is even, theorem (4.6).
Thus we get, for instance, the following
THEOREM 4.8: If a p-divisibZe design D=D(m,c,k,A' ,A) with a standavd automorphism group G having v' orbits on points and u orbits on cZasses exists then (i) If m is even then n=k2-vA=(k-A')+C(A'-A) is a square and either k-A' is aZso a square or v'=u (mod 2). (ii) If m is odd and c is even, then (k-A') is a square and either n is aZso a square or u is odd (iii) If m and c are both odd we have condition (28).
For divisible designs the previous results can be improved, as we can consider the quotient symmetric pseudodesign 2-(m,k+x'(c-l),cA) (see (4.4». The group G acts on it standardly and has
~
fixed "points" and u orbits.
Since
theorem (4.6) actually concerns matrices and not incidence structures, we can u-l apply it to this pseudodesign and we get that n also must be a square. Hence
396
D. Ghinelli Smit
we have THEOREM 4.9: fieaessaY'Y aondition foY' the existenae of a divisible design U=D(m,c,k,A',A) with a standard automoY'phism gY'oup of oY'der pOl:nts,
1>
having N fixed
fixed point alasses is that
(k-A')
(32)
~
v'-u
u-1 and n are both perfeat squares.
Hei'e v' and u are defined as before Y'elatively to the points, but it aan be shown that they, as well as 4> and N, have the same value relatively to the bloaks.
5.
THE HAW NONEXISTENCE THEOREMS
Consider a p-divisib1e design D=D(m,c,k,A',A) with a standard automorphism group G of order u. ( 1)
Let 4>,N,h=(m-4»/u, v' ,n as in (4.5) and let
l = (I h xC) e K + u AJ v I
,
where C is a (k-A ' ,A'-A,c)-matrix and K is defined in (21)4' We have seen in section 4 that the matrix Z is rationally congruent to the matrix S-l defined by (11)3 (see theorem (4.5»).
Hence a necessary condition for
the design to exist is that (2)
(-1,-1) (_1,~)N(N+1)/2
H (Z) = H (S-l) p
p
for every prime p. From lemma (2.1), we have (3)
Therefore, in order to calculate H (Z), we have to calculate H «I x C) e K) p P h (that is, by (1.6) Hp (C), Hp (K) and (iClh,IKI) and 1+ ~Ao«Ih x C-l) e K- l ». Applying the lemmas of section 2 we can show that (4)
-1 l+lJAo«Ih xC)
e
-1 2 K ) = kin.
2 From this, as k /n ~ n and III = ~-NIHI2 ~ ~N, we get (5)
Hp(Z) = Hp«Ih x C) e K) (-lJ
N+l
A,n).
Comparing (5) and (2) we can derive the following THEOREM 5.1: If a p-divisibZe design D=D(m,c,k,A',A) with a standard automoY'phism
397
On divisible designs
group G of order jl having N fixed points,
v'={(v-N)/jl}+N orbits on points and
1>
u={(m~
fixed classes, (and thus
)/]1
J~~
orbits on classes) exists, then
for every prime p
(6)
(k-A' ,(_1)(V'-u)(v'-u+l)/2 c U ]1N+1»p (n,(_1)U(U-l)/2 cUAjl¢+1)p = (n,_l)u-l. The proof (see [10]) is extremely tedious and long, but is rather straight-
forward, relying highly on the lemmas proved in section 2 and on (16)4-(18)4' This should be clear from the definition of K and from the properties (1.6)-(1.9) of the Hasse invariant.
We observe that if ]1 > c the proof is very simple and
could be given directly as then (7)
K = ]1(I /
Nc
x
C)
REI4ARK 5.2: (i) If]1 = 1 (=> v'=v, u=m, v'-u=m(c-l» (8)
then (6) gives, as (n,-l)
u-l
=1
(k_A',(_1)m(C-l){m(C-l)+l}/2 Cm)p (n,(_1)m(m-l)/2 cm A)p = 1
which, by (26)2' is exactly the condition in Bose and Connor [31, and combined ~/ith
(29)4 gives the well-known Bose and Connor Theorem. ( v'-u)!v'-u+1)/2 u N+q, (i i) If c= 1 (=> v=m, 1> =N, v' =u) the (-1) \ c i s a square u-1 and (6) becomes, as (n,-l) ='1, (9)
(n,(_1)v'(v'-1)/2 A]1 N+1)p = 1,
that is the condition given by Hughes in [16] , which, together with (30)4 gives the Hughes theorem. (iii) If ]1=c=l we have, of course, the second condition of the Bruck-RyserChow1a theorem. Hence, combining (4.6) and (5.1) we have a simultaneous generalization and, a 1so, a unifi ed proof of Bose ant Conn:)t', Hughes, and Bruck-Ryser-Chow1a Theorems. From (1,.3) we can rephrase the conditions in terms of Diophantine equations, and state the following nonexistence theorem. THEOREM 5.3: A necessary condition for the existence of a point-divisible design D=D{m,c,k,A',A) with a standard automorphism group G of order ]1 which fixes N points andq, classes (and so has v'={(v-N)/]1}+N orbits on points and
u={(m-1»/]1}+q, orbits on cZasses) is that (i) if v' and U both even, or m is even, then n is a square and the equation
398 ( 10)
D. Ghinelli Smit
(k-\') x2 + (_1)(V'-U)(V'-U+l)/2 CU ~N+~ y2
z
2
possesses a nontrivial solution in integers. (ii) if v' is even and u is odd, or m is odd and C is even, then k-\' is a square and the equation (11 )
n x2 + (1)u(u-l)/2 -
u
C
Hl Y2
z
\~
2
possesses a non trivial solution in integers. (iii) if v', u,m and C are all odd, then (10) has a nontrivial solution in
integers if and only if (11) has suah a solution. (iv) ifv',m and
( 12)
C
are odd but u is even then
(k-\')n is a square and thus
\ N+ 1 2 = z2 n x2 + (1 - ) ( v' -1 ) /2 ,,~y has a nontrivial solution in integers.
For divisible designs the nonexistence theorem (5.3) can be improved applying the condition (9) to the quotient symmetric pseudodesign 2-(m,k+\'(c-l), CA)
(see also theorem (4.9)). More precisely, then
( 13)
(k-\')
v'-u
andn
u-l
are both squares,
and (14 )
( n, ( - 1) u(u-l)/2
Hl)
CA~
p
. p = 1 f or every prlme
are necessary conditions for the existence.
Hence we can state the following
THEOREM 5.4: If a divisibiZe design D=D(m,c,k,\',\) exists with a standard automOI'phism group of order
~,
and if ~ , N, v·, u, n are defined
as
above, then we have
the aondition (13) and the Diophantine equations (10) and (11) are equivalent and both possess nontriv1:al solutions in integers.
Applications of the nonexistence theorems given here can be found in [10], especially in the case of semi symmetric designs. We conclude by pointing out that all the results in this paper concern standard tactical decompositions of p-divisible designs and we do not need to
On divisible designs
399
assume the decomposition to be the orbit decomposition of some automorphism group. ACKNOWLEDGEMENTS: The Author acknowledges with great pleasure the guidance of Prof. D.R. Hughes, her research supervisor at the University of London. She would also like to thank the British Council and the University of Rome for support during the time of this research.
Bl BL IOGRAPHY 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
R.E. Block, On the orbits of collineation groups, Math. Z., 96 (1967), 3349. R.C. Bose, Symmetric Group Divisible Oesigns with the dual property, J. Stat. planning Inf., 1 (1977),87-101. R.C. Bose and W.S. Connor, Combinatorial properties of group divisible incomplete block designs, Ann. Math. Statist., 23 (1952), 367-383. R.C. Bose and K.R. Nair, Partially balanced incomplete block designs, Sankhya, 4 (1939), 337-372. R. Brauer, On the connections between the ordinary and the modular characters of groups of finite order, Ann. Math., 42 (1941), 926-935. P. Dembowski, Verallgemeinerungen von Transitivitatsklassen endlicher projektiver Ebenen, Math. Z., 69 (1958), 59-89. P. Dembowski, Tactical decompositions of A-spaces, Proc. Colloq. Found. Geometry Utrecht 1959; Pergamon (1962), 15-23. P. Dembowski, Finite Geometries, Springer-Verlag, New York (1968). D. Ghinelli Smit, On semisymmetric designs, (Westfield College, University of London, 1980). D. Ghinelli Smit, Nonexistence theorems for automorphism groups of divisiblE square designs, (Ph. D. Thesis, University of London, to be submitted~ J.K. Goldhaber, A note concerning subspaces invariant under an inCidence matrix, J. Algebra, 7 (1967), 389-393. M. Hall Jr., Combinatorial theory, Blaisdell, Waltham (1967). G.H. Hardy and E.r~. Wright, An introduction to the theory of numbers, Oxford Clarendon Press, (1938). H. Hasse, tiber die Aquivalenz quadrati scher Formen in Korper der rationalen Zahalen, J. reine angew. Math., 152 (1923), 205-224. D.R. Hughes, Regular collineation groups, Proa. Amer. Math. Soa., 8 (1957), 165-168. D.R. Hughes, Collineations and generalized incidence matrices, Trans. Amer. Math. Soa., 86 (1957), 284-296. D.R. Hughes, Generalized incidence matrices over group algebras, Ill. J. Math., 1 (1957), 545-551. D.R. Hughes, Biplanes and semibiplanes, in Combinatorial Mathematias, Springer Lecture Notes 686 (1978), 55-58. B.W. Jones, The arithmetic theory of quadratic forms, MAA Carus Math. Monographs 10, John Wiley and Sons Distr. (1950).
400
20. 21. 22. 23. 24. 25. 26.
27. 28.
D. Ghinelli Smit
D. Jungnickel, On automorphism groups of divisible designs (to appear). C.W.H. Lam and S.E. Payne, Erratum to "Generalized relative difference sets", PI·oo. Amer. Math. Sao., 56 (1976), 392. E.S. Lander, Topics in algebraic coding theory, Ph. D. Thesis, University of Oxford, October 1980. H. Lenz, Quadratische Formen und Kollineationsgruppen, Aroh. Math., 13 (1962), 110-119. H. LUneburg, Gruppentheoretische Methoden in der Geometrie, Ein Berioht. Jahresber. Deutsohe Math. Ver., 70 (1967), 16-51. W. Mielants, On the nonexistence of a class of symmetric group divisible partial designs, J. Geom., 12 (1979), 89-98. H. Minkowski. Uber die Bedingungen. unter welchen zwei Quadratische Formen mit rationalen Koeffizienten ineinander rational transformiert werden konnen, J. reine angew. Math., 106 (1980). 5-26; see also Ges. Abh. I. Teubner. Berlin and Leipzig 1911, 219-239. C.J. Mitchell. On divisions and decompositions of l-designs, Ph. O. Thesis, University of London (1979). C.J. Mitchell, Group divisible designs with dual properties, (to appear in Disorete Math.).
29. 30. 31. 32. 33.
G. Pall, The arithmetical invariants of quadratic forms, BuZZ Amer. Math. Sao., 51 (1945), 185-197. E.T. Parker, On collineations of symmetric designs, Proa. Amer. Math. Sao., 8 (1957), 350-351. S.E. Payne, Generalized relative difference sets. Proa. Arner. Math. Sao., 25 (1970), 46-50. D. Raghavarao, Construction and combinatorial problems in design of experiments, John Wiley and Sons, Inc. New York (1971). P.R. Wild, On semibiplanes, Ph. D. Thesis, University of London (1980).
Westfield College (University of London) Kidderpore Avenue London NW3 7ST, U.K. permanent address: Istituto Matematico "G. Castelnuovo" Citta Universitaria 00100 Roma, Italy
401
Annals of Discrete Mathematics 18 (1983) 401-408 © North-Holland Publishing Company
SOME RESULTS ON PARTIAL STEINER QUADRUPLE SYSTEMS Mario Gionfriddo
ABSTRACT In this paper we prove some results concerning DMB PQSs having at least an element of degree seven and with m = 14 or 15 blocks.
In a previous paper we
have determined, to within isomorphism, all Dr·1B PQSs having 12, 14, 15 blocks and satisfying some particular conditions.
By means of the results contained in this
paper, we purpose to construct all DMB PQSs having 12, 14, 15 blocks.
1. INTRODUCTION A Partial Steiner QuadrupZe System (PQS) is a pair (P,q) where P is a finite non-empty set and q is a collection of 4-subsets (called bZocks) such that every 3-subset of Q is contained in at most one block of q.
Using graph theoretic ter-
minology, we will say that an element x of P has degree d(x)=h if x belongs to exactly h blocks of q.
Clearly Lpd(x)=4.lql. XE
We will call the degree-set (DS)
of a PQS (P,q) the n-up1e DS = [d(x), d(y) •••. ] where x,y, •• are the elements of P.
If there are r., elements of P having degree h., for i=1,2, •• ,s, we will write 1
DS=[(h l ) ,(h ) , ••• ,(h)] ,wherer +r 2+... +r =IPI. 2r l rl s rs s 2 then we will write (h')l=h .• 1 1
Ifr.=l,forsomei, 1
Two partial quadruple systems (P,q1) and (P,q2) are said to be disjoint and mutually balanced (DMB) if they have not blocks in common and any given triple of
distinct elements of P is contained in a block of ql if and only if it is contained in a block of q2'
It is easy to see for any two DMB PQS (P,q1) and (P,q2)
that Iql I = Iq21. A Steiner quadruple system (SQS) is a partial quadruple system such that every 3-subset of Q is contained in exactly one block of q. In a previous paper [4] we have determined, to within isomorphism, all DMB
402
M. Gionfriddo
PQSs having 12, 14, or 15 blocks (observe that OMS PQSs with 13 blocks do not exist, see (2)) and satisfying some particular conditions (blocks with at least two elements of degree 4 or blocks with one element of degree 7) along with their admissible degree-sets.
Our aim is to construct all OMS PQSs with 12, 14, 15
blocks and after, by means of these OMS PQSs, to find pairs of SQS(v) with qv-12, qv-14, qv-15 blocks in common, for qv = v(v-l)(v-2)/24. In this paper, we prove some results concerning OMS PQSs having at least an element of degree 7 and with m = 14 or 15 blocks.
In (4) we have constructed
a~~
OMS PQSs with 12, 14, or 15 blocks and such that there exists in them a block with at least two elements of degree 4.
Therefore, in what follows we shall always
consider OMS PQSs having blocks with at most one element of degree 4.
(P,ql) and
(P,q2) will be two OMS PQSs with P = {O,l, •• ,n-l} and Iqll = Iq21 = m.
Further,
for i=1,2, we will put Li = x
E
P, then JK(x,i) = {b
E
~q.
P3(b), where P3(b) = {X : Xc b, IXI = 3}. If q. : ~ E b}, further'" = {x E P : d(x) = r}. Finalr
1
ly, if x,y E P, x F y, we will write (x,y)
r
to indicate that {x,y} is contained in
exactly r blocks of ql or q2"
The following results on OMS PQSs are known (2):
1) for every x E P, d(x) > 4
further n > 8, m ;:. 8 ;
2) if k = MAX {d(x): x E P}, then m ;:. 2 k ; 3)
if d(x) > 4, then d(x) ;:. 6 ;
4) if m> 8, then m = 12 or m ;:. 14.
2. RESULTS From now on (P,ql) and (P,q2) will be two OMS PQSs with m = 14 or 15 blocks.
Further we shall always suppose that IM4
n
bl
~
1 for every bE ql
U
q2.
The results are call obtained to within isomorphism. LEMMA 1: Let (P,ql)' (P,q2) be two OMS PQSs with M = 14 or 15 that 1M4 n bl ~ 1 for every b E ql U q2. ;chat d(x) = 7 and
(x'Y)2~
then
{x,y,a,b} {x,y ,c ,d}
{x,c,f,h}
{x,e,f,g} E ql {x,e,b,d} {x,d,g,h}
{x,e,a,c}
b~ocks
and such
If there are two e~ements x,y E P such
403
Some results on partial Steiner quadruple systems
{x,y,a,c} {x,y,b,d} {x,e,a,b}
{x, h ,c, d}
{x,c,e,f} {x,d,e,g}E q2 {x,f,g,h}
PROOF: Let {x,y,a,b}, {x,y,c,d} E ql' {x,y,a,c}, {x,y,b,d} E q2'
If {U,x,a,c},
{V,x,b,d} E ql and {W,x,a,b}, {Z,x,c,d} E q2' then it is easy to see that it is not U = V
W = Z and that U '* V or W '* Z implies {U,V} n {W,Z}
that if U
V (resp, W = l) then {U,V} n {W,Z}
{U,V} n {W,Z}
a~,
~~,
let {W,x,i,j}E ql'
We wove
In fact, let U = V and let
Since W ~ Z implies {U,V} n {W,Z}
Necessarily {W,x,a,d} or {W,x,b,c} E ql'
~~,
~ ~,
it follows W = Z,
Let {(i,j),(i',j')}= {(a,d),(b,c)} and
Since (x,j)3 in ql' then {U,x,i,j}E q2'
Observe that
(since (x, i')3 in ql) {W,x, i' ,j' } E ql if and only if {U,x, i' ,j' } E q2'
Moreover,
if {W,x,i'}, {W,x,j'} are contained in two distinct blocks of ql and {U,x,i'}, {U,x,j'} are contained in two distinct blocks of q2' then we have (W,x)3 and (U,x)3'
In every case d(x)
Suppose U = W.
~
Necessarily, it follows that {U,V} n {W,Z}
7.
We have
{x,y,a,b}
{x,y ,a,c} {x,y,b,d}
{x,y,c,d} {U,x,a,c}
~ ~.
E q
1
{U,x,a,b}
{V,x,b,d}
E q
2
{Z,x,c,d}
where it is not V=U=Z.
Let Z
~
Suppose V ~ U.
U.
hence {x,c,Z,V} and {x,b,U,Z} E ql' Therefore, it is V = Z.
If V
~
Z, then {x,b,V,Z} E q2'
It follows {U,x,c,Z} E q2 with {d,X,C,Z}E q2'
Since (x,b)3 in ql' then {x,b,V,e} E q2'
Necessarily
{x,c,V,e} E ql and {x,c,U,e} E q2' with d(x) < 7
At this point, we can say that necessariZy U Z
h.
We have {x,y,a,c}
{x,y,a,b}
{x,y ,b,d}
{x,y,c,d} {x,a,c,e}
E q
1
{x,b,d,e}
{x,a,b,e}
E q
2
{x,c,d,h}
Hence, necessarily {x,c,f,h}
{x,c,e,f}
{x,d,g,h} E ql
{x,d,e,g} E q2
{x,e,f,g}
{x,f,g,h}
V
W F Z.
Let U
V
W
e,
404
M. Gionfriddo
The statement is so proved. 0
THEOREM 2.1: Let (P,q1)' (P,q2) be two OMS PQSs with m = 14 or 15 bZocks and such ~hat 1M4 Ii
bl "
1 fei' every bE q1
that d(x) = d(y) = 7 and I
IK( x, i)
Ii
IK(y, i)
IK(x,i)
U
Ii
q2'
IK(y,i)F0 (for i=1,2), then
F 2.
1
PROOF: Let x = 1, y = 2, and let II«l,i)
p =
{D, 1,2, •••
If there are two eZ-ements x,y E P suah
,n-1}.
Ii
IK(2,i)
1
2, for i
1,2,
From lel11lla 1, we have
{l
,4,6,7}
{1
,3,4,7}
(l
,3,5,7)
{l,5,6,O}
; 1,6 ,9 ,0 }
{1
(1,7,8,9) E ql
{l,6,7,9}E q2
{l,5,8,O}
{l,5,7,8}
n,2,3,4}
{l,2,3,5}
{l,2,5,6}
{l,2,4,6}
Observe that (2,7,8,9) E q2'
,8,9,0}
Otherwise, we have that {3,5,7}, {4,6,7},
U,8,9} are contained in exactly three blocks b. such that b. Ef 1«2,2), hence 1
IK(2,2)
Ii
contained in blocks b. E IK(2,1) , it follows 1
1M4 d(6)
Ii
~
1
1e
the SQS(q+l) as a subdesign.
CSQS(q+l) with q a prime power> then ther>e exists a
k
CSQS (q +1) fOI' a Zl k > O. In [Z), Cho gave a doubling construction for certain CSQS(v). CHO THEOREM 1:
If there exists a
exists a CSQS(Zv). a.t leaBt
CSQS (v) with v == Z or 10 (mod lZ) then ther>e
.'foreoveI' if the CSQS(v) compr>ises n orbits then there exists
Zn pail'<Jis;; distinct CSQS(Zv).
The basis of Cho's construction is as follows: (a) From each full orbit in the CSQS(v), choose any block, say {w,x,y,zl, 0
w< x < y
may be formed which is characteristic of the orbit from which the 4-block is drawn. ordered difference triple may be formed from a 3-b1ock.
Similarly a cyclically Then, since each 4-block
contains four 3-b1ocks, each difference quadruple gives rise to four difference triples.
If mlv, define an m-beheaded CSQS(v), denoted by CSQS(v,-m), as a col-
lection of 4-b1ock orbits with the property that any 3-block whose difference triple entries have a common factor vim does not occur in the system but all other 3-b1ocks occur exactly once.
We note that in this definition it is not in
fact necessary for v to be admissible.
However, the significance of the defini-
tion is that if there exists a CSQS(v,-m) and a CSQS(m) then there exists a CSQS(v) . The main result in [4] is: COLBOURN AND COLBOURN THEOREM 1: If there exists a CSQS(2v,-2m) with v
=m(mod
2)
then there exists a CSQS(4v,-4m).
The theorem is of great importance and has wide ranging applications which are dealt with in Section 4. Finally in this section we announce that we have recently obtained product constructions for cyclic Steiner quadruple systems which with an appropriate parameter and minimal modification yield the constructions of Cho and of Col bourn and Col bourn. In [12], the following results are proved: GRANNELL AND GRIGGS THEOREM 1: If there exists a CSQS(u) and a CSQS(v), the latter being composed entirely of fuZl orbits then there exists a CSQS(uv,-2u).
THEOREM 2: If there exists a CSQS(u) and a CSQS(v,-m) with v
=m(mod
2) and the
latter being composed entirely of fulZ orbits then there exists a CSQS(uv,-um).
THEOREM 3: If there exists a CSQS (u), a CSQS (2u) and a CSQS (v) wi th v = 2 lO(mod 12) then there exists a CSQS(uv).
0)"
M.J. Grannell and T.S. Griggs
412
THEOREM 4: If there exists a CSQS(u), a CSQS(2u) and a CSQS(v,-m) with v,m
=O(mod 2)
and tne latter being composed entirely of fuZl and half orbits then
there exists a CSQS(uv,-um).
Theorem 3 and 4 with u
2 in effect give Cho's and Col bourn and Colbourn's
constructions respectively. 3. R-CYClIC AND S-CYClIC STEINER QUADRUPLE SYSTEMS CSQS(v) which admit additional automorphisms are also of interest particularly those which are stabilized by the mapping i reverse Steiner quadruple systems.
~
v-i, i.e. which are also
We will call these R-cyclic Steiner quadruple
systems and denote them by RCSQS(v).
An easy argument shows that no RCSQS(v) may
contain a half orbit and hence that v
=2,4,10 or 20(mod
tion.
24) is a necessary condi-
It has been conjectured that this is also sufficient, but little work ap-
pears to have been done on RCSQS(v). Of more importance both from an historical point of view and because of their use with Colbourn and Colbourn's construction are symmetric cyclic or S-cyclic Steiner quadruple systems, denoted by SCSQS(v).
These are defined as
CSQ5(v) in which each orbit contributing to the system is stabilized by the mapping i
~
v-i.
The present authors [9) and Diener [7) independently obtained
the following results on the structure of SCSQS(v). GRANNELL AND GRIGGS/DIENER THEOREM 1: If there exists an SC5QS(v) then v 4n whel'e the prime factors of n are aU ~i
= 1 or
5(mod 12).
= 2n
or
Moreover, if m is even
mlv then the SCSQS(v) contains an SCSQS(m) as a subdesign. We conjecture that the necessary condition given in the theorem is also
sufficient.
Until recently, most known CSQS(v) were also S-cyc1ic.
However, a
non 5-cyc1ic C5Q5(26) is given (although with a misprint) in a paper by Fitting [8)
of 1915. Constructional techniques for 5CSQS(v) have been investigated in two papers
(16J, [17] by Kohler.
The basis for this work is as follows:
For a given v, not necessarily admissible, define a graph H"'(v) where the set of vertices
= {{x,y,z}:x,y,z
E
{1,2, ... v-3}\{v/2},x,y,z unequal, x+y+z
v}
and the set of
Some recent results on cyclic SQS's
413
edges is defined by the relation (which it is easily verified is both irreflexive and symmetric) that {x,y,z} is joined to {x' ,y' ,z'} if there is some ordering X,Y,l of x,y,z such that {x',y',z'} = {X,X+Y,v-2X-Y}.
v
=2,4,10 or
In the case where
20(mod 24) careful analysis of the difference quadruples and the
difference triples they contain leads to the graph H'\(v) where each vertex reppresents two difference triples and each edge represents a difference quadruple. Hence KOHLER THEOREM 1: There exists an SCSQS(v) iff H*(v) aontains a l-faator. (This result is in fact slightly stronger than that given originally by Kohler [16] and ·is due to Diener [7].
The basic idea of the work goes back to
Fitting [8]). Kohler proceeds to prove various properties of the graph H*(v). ular it is not necessarily connected.
In the case where v
or 5(mod 12), H*(v) consists of two components.
In partic-
= 2p with p a prime
=
In one of these components,
H *(V), the elements of each set which comprises a vertex are all even and in the 2 other component, H,*(v), at least one element (and therefore precisely two elements) of each set comprising a vertex are odd.
Now it can be verified that a
l-factor of H,*(v) may be obtained by selecting the set of difference quadruples
< a,b,a,v-{2a+b)> with a
=
1,3, ... (v-8)/2 and b
=
2,4, ... (v-2a-4)/2.
In addi-
tion the graph H *(V), which is also denoted by H(v) in [16], can easily be seen 2 to be isomorphic to the graph H*(p) which leads to: THEOREM 2: There exists an SCSQS(2p) with p a prime
= 1 or
S(mod 12) iff H*(p)
contains a l-faator.
When p group < i
~
=5(mod
12), Kohler proceeds further with the analysis.
ai+b(mod p) > with a,b
E
GF(p), a
f
By using the
0, in order to reduce the number
of orbits required, another graph B(p) is obtained as follows: Let F = GF(p)\{0,1,(p-l)/2, p-2, p-l}.
If
a E
F define
= {a,l/a,-a/{a+l),-a-l,-l/a-l,a/(a+l)-l}. Then it follows that for a,S E F either ~ = $ or ~ n S =~. Take as the set of vertices of B(p), the set ~
{~ a
: a
= 8+1
E
F} and join ~ and or
a
= 8-1.
e by
an edge if there exists
a E ~
The following two theorems are proved.
and S E $ with
414
M.J. Grannell and T.S. Griggs
THEOREM 3: If Sip) co"Ztains a l-factoT' then theT'e exists an SCSQS(2p). THEOREM 4: If Sip) is oT'idgeless with p
= 53 OT'
77(mod 120) then theT'e exists an
SCSQ5(2p) . These results enabled Kohler to construct SCSQS(v) for v = 26,34,50,58,74, 82,106,178,202,226,274,298,346,394,466,586 and 634.
Using similar techniques Cho
[3) has constructed SCSQS{v) for other values of v not necessarily covered by the theorems. Enumeration results for R-cyclic and S-cyclic Steiner quadruple systems are few.
The unique CSQS(lO) is also S-cyclic.
For v
= 20, there exist 16 distinct
RCSQS(v) which partition into 4 isomorphism classes (Phelps [20]).
Of these, 4,
all within a single isomorphism class, are also S-cycl ic (Jain [15] ). authors [11) enumerated SCSQS(26).
The present
There are 87 distinct SCSQS(26) in 18
isomorphism classes. 4. EXISTENCE RESULTS FOR CSQS(v) In this section we apply the known methods of construction, particularly those dev; sed by Cho, Col bourn and Col bourn, and ourselves, to vari ous "small" systems constructed by hand or with the aid of a computer search to give an account of the known spectrum of CSQS(v). (a) In [4], Colbourn and Col bourn give a CSQS{16,-8).
Together with a
CSQS(32), two examples of which were constructed by the present authors [ 10) , all n
CSQS(2 ) for n
~
5 may be obtained using the following constructional scheme.
The
details appear in [4). CSQS(16,-8)--~--
C5Q5(32,-16)
~
CSQS/64,-32)
CSQS/128,-64)
-~-
CSQS(32)---+-- CSQS(64)---- CSQS(128}------n THEOREM A: CSQS(2 ) exist foT' n = 2 and n ~ 5.
CSQS(2 3 ) and CSQS(2 4 ) do not
(b) Colbourn and Phelps [5] constructed a CSQS(40,-20).
Again, together
with one of the known CSQS(20}'s a similar constructional scheme to the one above may be adopted.
Some recent results on cyclic SQS's
CSQSt40,-20)
-~--
CSQS!80,-40)
415
-~- CSQSf160,-80)--~
csQs(2o)-------csQs(4o)---------- CSQS(80)-------- CSQS(160)-------n THEOREM B: CSQS (2 . 5) exist for n;;' 1 . (c) It is clear that what is really required are appropriate "starter" systems to which Colbourn and Colbourn's construction may be applied. provided by the S-cyclic Steiner quadruple systems.
These are
From the theorem on the
structure of these systems, if m is even and mlv then by removing all those orbits which contribute blocks of the SCSQS(m) subdesign from the SCSQS(v), a CSQS(v,-m) is obtained.
In particular if m = 10 and u = vim is odd we have the
following: SCSQS(lOu)-----CSQ1(lOu,-10)-----CSQSf20u'-20)--~--CSQSf40u'-40)-~-CSQ~ (lOu)
CSQS (20u)
CSQ5(lO) CSQJ(20) n using the CSQS(2 .5) from the·previous theorem.
CSQSI( 40u) CSQst(40) An SCSQS(50) is given by Kohler
[16], and Cho [3] has constructed SCSQS(lOu) for u
=
13,17,25,29 and 37.
n THEOREM C: CSQS(2 .5u) exist for n;;' 1 and u = 5,13,17,5 2,29 and 37. (d) The results given in Theorems A, Band C appear to be the only known infinite families, the orders of whose members increase by a factor of 2, which are totally determined. In many others there are gaps. Using our Theorem 3, let u = 2nw for some n ;;. 1 and w = 5 or 5u for any of the values of u given in Theorem Then for any v = 2x, with x odd, for which a CSQS(v) exists, it follows that a CSQS(2 n+ l wx) exists i.e. only the existence of a CSQS(2wx) is left undetermined. C.
Values of x for which CSQS(2x) are known are x = 5,11,13,17,19 (Colbourn and Phelps [51),5 2 ,29,37,41 and 72 (Cho [31) as well as other values greater than 50 constructed by Kohler [161 and listed towards the end of Section 3. THEOREM D: If there exists a CSQS(2x) with x odd then there exist CSQS(2 n.5x) and n
2
CSQS(2 .5ux) for u = 5,13,17,5 ,29 and 37 and n;;' 2. (e) The existence of a CSQS(2x) with x odd also implies the existence of another infinite family.
Immediately from Cho's construction there exists a
Some recent results on cyclic SQS's
416
CSQS(4x) and also, by removing the quarter orbit, a CSQS(4x,-4).
Applying n n Colbourn and Colbourn's construction one can then produce CSQS(2 x,-2 ) for all n n ~ 2 and when n ~ 5 these systems can be completed using the CSQS(2 ) from Theorem A. Again a gap appears i.e. the existence of a CSQS(2 3x) and a CSQS(2 4x) is left unresolved. THEOREM E: If there exists a CSQS(2x} tJith x odd then there exists CSQS(2 n.x) for n
=2
and n
~
5.
(f) There are other values of v for which CSQS(v) are known to exist. These include v = 2n.7 for n = 2 and n ~ 5, constructed as in (e) from a CSOS(28). Phelps' theorems enable further systems to be constructed and there are applications of our Theorem 3 in addition to that given above such as the construction of n
a CSQS(2 .5xy) for n ~ 3 from a CSQS(2x) and CSQS(2y) with x and y odd. [3J has constructed CSQS(v) for v
=
Also Cho
88,92 and 124.
5. CONCLUDING REMARKS The evidence of the results in the previous section would seem to support the conjecture that CSQS(v) exist for all admissible v apart from v = 8,14 and 16. The only values less than 100 for which the existence of a CSQS(v) is unresolved are v
= 46,56,62,70,86 and 94. A major problem with the methods known at present appears to be that the
non-existence of CSQS(8), CSQS(14) and CSQS(16) frustrate the construction of certain systems of higher order.
To overcome this, it would be useful to have a
generalisation of Cho's construction which in effect "doubles" the quarter orbit. If v
= 4m then starting from a CSQS(4m,-4), a CSQS(8m,-8) can be constructed but
not completed.
However, it seems not unlikely that the removal of a relatively
small number of orbits from the CSQS(8m,-8) may result in completion being possible. Other major lines of investigation would appear to be the construction of CSQS(2x) with x odd and the development of more general recursive constructions. Finally, the question of whether the known necessary conditions for R-cyclic and S-cyclic Steiner quadruple systems are also sufficient remains open.
M.J. Grannell and T.S. Griggs
417
BIBLIOGRAPHY 1.
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
J.A. Barrau, Over de combinatorische opgave van Steiner, Kon. Akad. Wetensah. Amst. Verlag Wis-en Natuurk. Afd., 17 (1908), 318-326. (= On the combinatory problem of Steiner, Kon. Akad. Wetensah. Amst. Proa. Seat. Sai., 11 (1908), 352-360). C.J. Cho, On cyclic Steiner quadruple systems, Ars Combinatoria, 10 (1980), 123-130. C.J. Cho, private communication. C.J. Col bourn and M.J. Co1bourn, A recursive construction for infinite fami1es of cyclic SQS, Ars Combinatoria, 10 (1980), 95-102. C.J. Co1bourn and K.T. Phelps, Three new Steiner quadruple systems, Utilitas Mathematiaa, 18 (1980), 35-40. 1. Diener, On cyclic Steiner systems S(3,4,22), Annals of Disarete Math., 7 (1980), 301-313. I. Diener, On S-cyc1ic Steiner systems, unpublished. F. Fitting, Zyk1ische Losungen des Steiner'schen Problems, Nieuw Arah. wisk~ (2) 11 (1915), 140-148. M.J. Granne11 and T.S. Griggs, On the structure of S-cyc1ic Steiner quadruple systems, Ars Combinatoria, 9 (1980), 51-58. M.J. Granne11 and T.S. Griggs, A cyclic Steiner quadruple system of order 32, Disarete Math., 38 (1982), 109-111. M.J. Granne11 and T.S. Griggs, An enumeration of S-cyclic SQS(26), Utilitas Mathematiaa, 20 (1981), 249-259. M.J. Grannel1 and T.S. Griggs, Product constructions for cyclic block designs -I, Steiner quadruple systems, (to appear). M. Guregova and A. Rosa, Using the computer to investigate cyclic Steiner quadruple systems, Math. Casopis Sloven. Akad. Vied., 18 (1968), 229239. H. Hanani, On quadruple systems, Canad. J. Math., 12 (1960), 145-157. R.K. Jain, On cyclic Steiner quadruple systems, M. Sc. thesis, McMaster University, Hamilton, Ontario (1971). E. Kohler, Zyk1ische Quadrupe1systeme, Abh. Math. Sem. Hamburg, 48 (1979), 1-24. E. Kohler, Numerische Existenzkriterien in der Kombinatorik, Numerische Methoden bei graphentheoretischen und kombinatorischen Problemen, Birkhauser, Basel (1975), 99-108. C.C. Lindner and A. Rosa, Steiner quadruple systems - a survey, Disarete Math., 21 (1978), 147-181. K.T. Phelps, Infinite classes of cyclic Steiner quadruple systems, Annals of Disarete Math., 8 (1980), 177-181. K.T. Phelps, On cyclic Steiner systems S(3,4,20), Annals of Disarete Math., 7 (1980), 277-300.
Division of Mathematics and Statistics, Preston Polytechnic, Corporation Street, Preston, Lancashire, PR1 2TQ. ENGLAND
This Page Intentionally Left Blank
Annals of Discrete Mathematics 18 (1983) 419-426
419
© North-Holland Publishing Company
ON A CONJECTURE OF S. ILKKA L. Guerra and E. Ughi
,'t
I NTRODUCTI ON In this paper we give a new proof of a well known conjecture of S. I1kka (1972) [31, which concerns regular and pseudoregu1ar points with respect to an affine conic C defined over a finite field F , and which claims that such points, q
apart from the centre of the conic, may only exist for small values of q.
This
conjecture was demonstrated in 1973 by B. Segre [71 albeit through a very laborious and detailed study. In a recent work, we were concerned with the distribution of squares and non-squares in a Galois field, emphasizing that it could have useful applications to combinatorial geometry.
The aim of this paper is to show a much shorter proof
of the conjecture of S. I1kka than B. Segre's may be obtained, as one of these applications.
Substantially, this proof relies only on the Hasse-Weil theorem on
the number of points of a curve defined over a finite field. We refer to the work of B. Segre [71 for everything concerning general properties and definitions and for a detailed description of the exceptional cases. There exists another exceptional case pointed out by F. Karteszi [4]; see also the last sentence of
§
2.
After a section containing preliminary material, in the following two sections we give our proof of the conjecture of S. I1kka separating the case of an ellipse from that of a hyperbola. In the latter case, we obtain explicit
t
formulas for the number of regular and pseudoregu1ar secants through a point P not belonging to the hyperbola (and different from the centre), relating those numbers to the number of points of an elliptic plane curve defined over F. q
We hope
that these formulas will allow us to answer in a future paper, at least in this case, the general question proposed by B. Segre at the end of the introduction to his fundamental paper. We extend our heartiest thanks to prof. G. Korchmaros for his valuable help
420
L. Gue"a and E. Ughi
during this work.
1. PRELIMINARY Let Fq be a Galois field of order q, which is assumed to be always odd in this paper; let
be an affine plane over Fq ; let C be a non singular conic of 11. Then, it is well known that C contains exactly q+l pOints of 11, no three points of 11
which lie on a line (and which include possible points at infinity); moreover the points of are
11
exterr~Z
not belonging to C fall into two classes, according to whether they or internal with respect to C.
This distribution may be related to the concept of an internal or external point with respect to a segment of line AB (A,B
E 11,
A ~ B).
Indeed, recalling that a point P belonging to the line AB is said to be external or internal to the segment AB according to whether the cross-ratio (A B P (AB),,,) is a square or a non-square if Fq ( (AB)oo = the point at infinity of the line AB), then we have (cf. [7], LEMMA 1: If A,B
E
§
10):
C, A ~ B, then the set of the (q-l)/2 external (internal) points
to the segment AB of the line AB is the set either of the external or of the internal points with l'espect to the conic C (including (AB)oo).
In other words, for any secant line AB, either the two classes of the external and the internal points remain fixed or they are exchanged, according to whether the point at infinity (AB)oo is external or internal with respect to the conic C. This allows us to divide the affine secants to C into two classes, corresponding to each one of the two given possibilities; we call any secant line which does not exchange the character of points regular, and any other we call pseudoregular.
Naturally, there arises at this point the problem of classifying any point pE
11,
P ~ C, according to the behaviour of the secants to C through P.
These
lines are divided into two classes, that of regular and that of pseudoregular ones, whose numbers of elements we call respectively Rp' Sp.
On a conjecture of S. Ilkka
421
As we pointed out in the introduction, the questions of calculating the integers Rp' Sp for any point PE IT and of describing the outline of regular and pseudoregular secants through P or, more generally, the configuration of the set of points P E
having given values of the characters Rp' SP' are even today still But, if we confine ourselves only to the extreme cases Rp=O, Sp=O,
open ones.
IT
then the question may be answered. Call any point P for which Sp=O regular, and any P for which Rp=O pseudoregular.
Then it can be proved that:
CONJECTURE OF SEPPO ILKKA: if the conic C has a centre, then regular or pseudoregular points, different from the centre, may only exist for a finite number of values of q.
We shall give our proof of the conjecture in the following sections.
For
the moment we point out that the case of a parabola is exceptional, because of the following (cf. [7),
§ 8}:
LEMMA 2: Any secant to a parabola C of
TI
is regular.
Moreover, the centre of C , when C is an ellipse or a hyperbola, is always a regular or a pseudoregular point, according to the following (cf. [7),
§ 14):
LEMMA 3: The centre of an ellipse is a regular or a pseudoregular point according to whether - 1 =f}.or - 1 =0, i.e. q == - 1 (4) or q == 1 (4); while the centre of a hyperbola is a regular or a pseudoregular point, according to whether - 1
=C
or
- 1 = /)..
(0 denotes, as usual, a non zero square in Fq , and /). a non-square in Fq}. NOTE: We emphasize that our definition of a regular or a pseudoregular point is slightly different from that of [7), as we prefer not to include the limiting case of a tangent line. We end this section with some algebraic preliminaries concerning the distribution of squares and non-squares in Fq and the Hasse-Weil theorem.
L. Guerra and E. Ugh;
422
As usual, (~) is the Legendre symbol of x in F , which equals x(q-l)/2, so q
q
that it is 0,1-1, according to whether x is respectively zero, a square or a nonsquare in F. q
The key result is the following:
LEMMA 4: :'et t'zere be given r polynomiaZs
fl(X), ... ,fr(X)
FJ X], of positive
E
degree, with no multiple zevos, and such that no two have a common zero.
Then
"i:ere exists an integer C, depending only on the degvees of the polynomials fi(X), and such that, for any fixed r-tuple of quadratic chal'actevs £i = ± 1, i =1, ... , r ,
Ci~X) )
=
E.
1
c.Z1J:J.ys has a solution in F , fov q ;;. C.
q
PROOF: It is clear that the question depends only on the case
Ci~X))= -
Indeed, for any equation
1 we may substitute
1, i = l, ... ,r.
E. 1
(gi~X))
1, where
g.(X) = s f.(X), s being a non-square in Fq , which is equivalent to the former, so 1 1 that the general assertion that holds in Fq follows from the case i = 1, ... ,r.
= 1,
Now, in this case, call the number of solutions of the given system
of equations N.
N=
E. 1
Clearly we have, (cf. [ 2),
>JFq [1
+
Cl~X))]
§ 1)
[1 + C~(X))] +
N ,
o
where, if we denote by Z (f.) the set of zeros of f.(X) in F , then q
1
1
q
Distributing the various terms and collecting those which contain the same number of factors fi' we obtain N
.9.
2r
+
2r
r i ~1
[J,J:(X)) l
1 L Cl:X') ... C<X)) + N0 . 2r xEF q
+-
Then we can write
On a conjecture of S. Ilkka
423
where Ih
't'f
= lEF E
q
(f..(g) q
and f is one of those divisors of the polynomial fl(X) ... fr(X) obtained by multiplying the various factors f.(X). 1 Now observe that, denoting the set of zeros of f. in the algebraic closure 1
of Fq by Z(f.), then 1
I
N
a
1
I.;;; -2 I
r
.u
1=1
Moreover, repeating the arguments used in [2], lemma 1.5, one can easily prove that
2}
and consider the map of the set consisting of all pairs (g,h) of two different elements of U into
defined by
I
(g,h)l-.... g
n
h.
Here each element of 'ie and AS has (q-l)(q-2) preimages.
If ISBI = 6(q+l)2/v ,
then the number of preimages of B is (3q+3)(3q+2) and IBSI = ~ q3(q2_ q+l )(q_l). 3
3
3
But then (q +l)(q-l)[ (q +l)(q-l)-ll=lul(IUI-l»(q +1)(q-l)(q-2)+
.~ q3(q3+ 1)(q_l)(q_2)+ -
2
1\1
S
i
q3(q2_q+l )(q-l)(3q+3)(3q+2), which is impossible. 1
3
2
Thus
= 2(q+l) lv, IB 1="2 q (q -q+l)(q-1) and I[BJ n ul= q+1. As IUI(lul -1) = 3 133 132 = (q +1)(q-l)(q-2)+ 2 q (q +l)(q-l)(q-2) +"2 q (q -q+l)(q-l)(q+l)q, we see that 1 = Q U AS U BS.
e
431
On homologies of fixed-paint-free affine groups
As
I
Let g
E
Finally, we can investigate the homology n. point X E Re which is not left invariant by n. As Xn
+ X,
gn
+ g,
so that Y E I.
In fact, YE 1\
Q
e
Rei> 2, there exists a [X] n U and Y = g n gn.
= AS
U
BS, and we can assume
that Y E {A,B}. Clearly, n leaves invariant
U
ry n
[y] n U
and hence the set of axes
a] n U.
yE[Y] r)U
But this set generates the desarguesian subplane (P,B).
Therefore n induces a
homology of (F,B) leaving invariant the dual ovalO, which clearly is impossible.
BIBLIOGRAPHY 1. 2. 3. 4.
5. 6.
7.
8. 9.
J. Andre, Uber Perspektivitaten in endlichen projektiven Ebenen, A~ch. Math., 6 (1955), 29-32. A. Barlotti, Le possibili configurazioni del sistema delle coppie punto-retta (A,a) per cui un piano grafico risulta (A,a)-transitivo, BoZZ. Un. Mat. ItaZ., 12 (1957), 212-226. R.W. Hartley, Determination of the ternary collineation groups whose coefficients lie in GF(2n), Ann. Math., 27 (1926), 140-158. C. Hering, On projective planes of type VI. Teorie Combinatorie, Torno II, Rome, 1976. (Coll. Intern. Teorie Combinatorie, Accad. Naz. dei Lincei, Roma 1973). Atti dei Convegni Lincei 17, Torno II, (1976), 29-53. C. Hering, On the structure of finite collineation groups of projective planes, Abh. Math. Sern. Univ. Hamburg, 49 (1979), 155-182. C. Hering, Finite collineation groups of projective planes containing nontrivial perspectivities. The Santa Cruz Conference on finite groups, Santa Cruz 1979. Proceedings of Symposia in Pure Mathematics 37 (1980), 473-477. C. Hering, On shears in fixed-poi nt-free affine groups. Finite geometries and designs. Proceedings of the Second Isle of Thorns Conference 1980, edited by P.J. Cameron, J.W.P. Hirschfeld and D.R. Hughes, London Math. Soc. Lecture Notes 49, Cambridge 1981, pp. 146-152. C. Hering, On Beweglichkeit in affine planes. To appear. C. Hering and M. Walker, Perspectivities in irreducible collineation groups of projective planes, II. J. Statist. PZann. Infe~ence, 3 (1979), 151-177.
432
10. 11.
C.Hering
B. Huppert, l:'16.Uche GT'uppen 1. Berlin, Heidelberg, New York, 1967. H. Lenz, Kleiner desarguesscher Satz und Dualitat in projektiven Ebenen, 3 the above values of m(4,3) and m(5,3) are the only known values of m(r,q). Two k-caps are said to be equivaZent if there is a linear transformation mapping one into the other.
Up to equivalence, the ovaloids in S2,3 and S3,3 are
unique, being respectively conics and elliptic quadrics, while the uniqueness of the 56-cap in S5,3 was shown in [4]. In [6], Pellegrino showed that every 20-cap in S4,3 is one of two geometric types, but it is by no means clear how many inequivalent 20-caps there are of each type.
Our main aim is to show that there are in all exactly nine inequivalent 20-
caps in S4,3'
We will also show that just two of these occur as the intersection
of a 56-cap in S5,3 with a hyperplane. 2. NOTATION AND DEFINITIONS We write pOints of Sr,q as (r+l)-tuples (a l ,a 2,·· .,a r +l ), where a i £ GF(q) and not all the ai's are zero; (a ,a 2,· .. ,a + ) = (b l ,b , ... ,b + ) if and only if l 2 r l r l Ab. for all i, for some A£GF(q) with A f O. 1
A line in S which meets a cap K in exactly two points is called a secant r,q
434
R. Hill
of K.
For any point P of S not in K the degree of P with respect to K is the r,q number of secants of K passing through P. The intersection of a cap K with a
hyperplane H of Sr,q is called an i-section of K if IK n HI = i. A m;;,ri.r of a k-cap K in Sr,q is a k x (r+l) matrix over GF(q) whose rows are the points of K. Clearly, any permutation of the rows, or multiplication of a row by a non-zero scalar, yields a matrix of the same cap. Let G denote the group of non-singular linear transformations of S r,q r,q We regard the elements of Gr,q as non-singular (r+l) x (r+l) matrices which transform points of Sr,q via right multiplication. Two caps Kl and K2 are said to be
.:ql
2 ;
20
56
When n = 2, a k-cap is usually called a k-ara. For q odd, a (q+l)-arc is a conic and a (q2+1)-cap is an elliptic quadric. The following three theorems were shown by Segre [4] in 1967. THEOREM 1.1: For q odd, a k-ara in PG(2,q) with k> q unique aonia.
t Iq + ~s aontained in a
0
THEOREM 1.2: For q odd and sUffiaientZy Zarge, a k-aap in PG(3,q) with 2 3/2 1 k > q - cq , where c < 4' is aontained in a unique eZZiptia quadria. THEOREM 1.3: For q odd and suffiaientZy Zarge, m2(n,q) < qn-l - cq n-3/2 where c
2 and n > 4,
q
45\
Caps in elliptic quadrics
m2(n,q)
~
n-4 n-4 n-5 q m2(4,q) - q - 2(q + ... + 1) + 1.0
The estimate of m2(4,q) from Theorem 1.4, namely that, for q even and q > 2, 3 1 3/2 3 + '4 + 1 , m2(4,q) < q - 2" q gave the following improvement on Theorem 1.4 for all even q > 2.
These values
are not as good for those q for which Theorem 1.5 applies. THEOREM 1.7: For q even, q> 2 and n
~
5,
n-l 1 n-5/2 3 n-3 n-5 m2(n,q) < q - 2" q + '4 q - 2(q + ... + 1) + 1. 0 There are results due to Barlotti [11 comparable to Theorem 2.10 which are effective for small q. THEOREM 1.8: Let K be a k-cap in PG(3,q), q odd. and k ~ q2 for q
= 3 or 5,
If k
~ q2 - q
+
7 for q ~ 7
then K is contained in a unique elliptic quadric.
For further results of this type, see [11, [41, [51.
0
For a list of previous
estimates for m2(n,q), see [41, p. 166. The results and proofs in the next two sections are either the same or just slight variations on those in [41.
However, the nature of the results is such
that, without the full details, it would be difficult to follow the argument. The following notation is also used: y y
+
= GF(q), GF (q)
U {oo}
P(X) = the paint of PG(n,q) with coordinate vector X = (xO""'x n) , V(F 1' ... , Fr) = {P (X) E PG (n , q) I F1(X) ITd = subspace of dimension d.
F (X) r
O},
452
J. W.P. Hirschfeld
2. ELLIPTIC QUADRICS IN PG(3,q) For notation and background material on PG(n,q), we follow [3].
The irre-
ducible quadrics in PG(n,q) are En
= V(f(xO'x l )
+ x2x3 + •• + xn_lx n)
1
I
Hn = V(XOX l + X2X3 + ••• + xn_lx n)
n odd;
n even; Then every quadric Q in PG(n,q) can be
f is an irreducible binary quadratic form.
written ITrV, where llr is the subspace of singular points and V is a section of Q 1 skew to II ; that is, Q is a cone with vertex llr and base n-rr ducible plane quadric is a conic. by a
Jl
V.
An irre-
LEMMA 2.1: In PG(3,q). the maximum value of k for which a k-cap can lie on a quadric sUl'face is as follows.
Symbol
q even
q odd
repeated plane
IIlo
q +2
q +1
1i ne
III El
2
2
plane pair
2(q + 2)
2(q + 1 )
2(q + 1)
2(q + 1)
hyperbolic quadric
1l1Hl 1l 0P2 H3
elliptic quadric
E3
2(q + 1 ) 2 q +1
2(q + 1) 2 q +
Description
cone
LEMMA 2.2: In PG(3,q), q> 2, Then, if k;.
t
0
let K be a k-cap contained in a (q2 + l)-cap O.
(q2 + q + 4),any cap K' containing K lies entirely in O.
PROOF: Suppose there is a point Q in K' \ O. meets K in at most one point.
Each of the q+l tangents from Q to 0
Hence there are at least k - (q+l) ;.
21
(q
2
- q + 2)
lines through Q which meet K and are bisecants of O. So 0 contains at least 122 (q + 1) + 2 • 2 (q - q + 2) = q + 3 points, a contradiction. 0 LEMMA 2.3: In PG(3,q), let K be a k-cap in a quadric V.
Then, if
453
Caps in elliptic quadrics
k~
21
(q + q + 4)
2
k ~
21
(q + q + 6)
2
for q
~
5 ,
for q
= 3,4,
V is elliptic and every cap containing K lies on V.
PROOF: This follows immediately from Lemmas 2.1 and 2.2.
0
LEMMA 2.4: If A,B,C are three sets such that C ~ A U B,then IA n BI
~
IAI + IBI -
lei.
PROOF: This follows from the identity IA n BI
IAI + IBI - IA U BI. 0
LEMMA 2.5: If PG(3,q) with q odd, let C and C' be conics in the respective planes 11
and
11'
such that the line R. =
11
n
11'
meets both C and C' in the points It and P'.
Then the quadrics containing C and C' form a pencil F if n2 ,n ,n
+
-
= {Fp I
p
E y+}
such that,
3 4 and n4 are respectively the number of plane pairs, cones, hyperbolic
quadrics and elliptic quadrics in F, then
n2
= 1 • n3 = 0 or 2 ,
+
n 4
(q-l-n )/2 3
~
(q-3)/2
n4
(q+l-n 3 )/2
~
(q-l)/2
The plane pail' in F is nal to C is internal
PROOF: Let
11
11
01'
= V(x ) and 3
+ 11', and
n3 is 0
2 as a point P on
01'
R.
which is exter-
external to C'.
11'
= V(x 2 ); let P = P(l,O,O,O) and Q = P(O,l,D,D). Also
2 let C = V(X 3 ,X/ - xOx l ) = {P(t ,1,t,O) I tE /}; then C'=V(x ,bxOX + cx x + 2 3 l 3 + dX/ - xOx l ) = {P(cs+d,s2 - bs,O,s-b) I s E /L Hence, if Fp = V(Fp) is a ,
.
quadric containing C and C • then for some p ln F
p
y
+
,
J. W.P. Hirschfeld
454
So, with F( i )
aF
P
fjx. 1
-x
1
+ bx , F(l)
3
So F is a plane pair when p = '" and otherwise has a singular point where F(O)D= F(l) = F(Z) = F(3) = 0, which occurs at P(2c,2b,-p,2) when p Z = 4(bc + d). Hence n = 1, and n3 = 2 or 0 as bc + d is a square or not. Also 2 P(l,-r,O,O) is external or internal to C as r is a square or not, and is external or internal to (' as (bc + d)r is a square or not. +
-
It remains to find n and n . If Q and Q' denote respective points of ( 4 4 and (' other than P and P', there are (q_1)Z lines QQ', each of which lies on exactly one F. p
Also, every line on an irreducible Fp not through P or P' is some
QQ'.
Each cone contains q-1 lines QQ' and each hyperbolic quadric 2(q-l) lines
QQ'.
So +
n3( q- 1) + 2n 4 ( q- 1) = (q - 1) +
n3 + n4 + n4
=q
Z
,
,
whence the result. 0 Three conics (1,C -cetl"ali<JLirai:
2
and C in the respective planes 3
IT
l ,IT2 and
IT3
form a
system if
with P,P ,P ,P3 not coplanar. 1 2 The tetrahedral system is frat if the tangents
~1'~2
and
~3
to Cl 'C and C3 at P 2
are coplanar. LEMMA 2.6: The three conics of a tetrahedral system are contained in a quadric if and ,m"y if the system is [rat: the quadric is then unique and irreducible.
PROOF: If. in the above notation, the conics form a tetrahedral system, then by
455
Caps in elliptic quadrics
the previous lemma there is a pencil F of quadrics Fp through C and c • If the l 2 system is flat, each Fp has the tangent plane ~1~2~3 at P. So such an Fp is tangent to C at P and contains the points P and P2 of C • Hence a quadric F 3 l 3 p containing one further point of C3 contains C and is necessarily irreducible. As 3 two quadrics cannot intersect in a curve of degree more than four, the quadric is unique. If there is a quadric Fp containing the conics Cl 'C and C of a tetrahedral 2 3 system, then Fp has a simple point at P = Cl n C2 n C3 • So the tangent plane TI at P to Fp meets TI 1 ,TI 2 and TI3 in the tangents £1 and ~3 are coplanar. 0 LE~~A
'~2
and £3 to the conics; so
tl'~2
1
2.7: In PG(3,q) with q> 13, let K. be a k.-arc with k. > q - -4 Iq + 7/4 and 1
so contained in a conic C , i
i
= 1,2,3.
1
1
If C ,C and C form a non-flat tetrahe-
l
2
3
dral system, then there is a line meeting K ,K and K3 in distinct points. l 2 Kl U K2 u K3 is not a cap.
So
PROOF: By Lemma 2.5 there is a pencil F of quadrics Fp through C and C . 1 2 such Fp has in common with C3 the points P'P l and P2 ' where, as above,
Any
C. n C. = {P,P }. As the system is not flat, there is exactly one F tangent to k 1 J p C3 at P, one Fp tangent to C3 at P and one Fp tangent to C3 at P2• As there are l n + ~ (q-3)/2 hyperbolic quadrics in F, at least (q-3)/2-3 = (q-9)/2 hyperbolic 4 quadrics in F meet C in a point Q other than P'P and P • Let M be the set of 3 l 2 such points Q: then m = IMI ~ (q-9)/2. 1
P } and put r i = 1Ki' I; then r3 > q - "4 Iq - 5/4 and, li 2 for i = 1 and 2, r i > q - "4 Iq - 1/4. Also, let M' = M n K3' and m' = 1M' I; finally, put C ' = C\{P,P ,P } and so IC 'I = q - 2. Then, by Lemma 2.4 with 3 l 2 3 3 Let Ki'
A
= M,
B
= K3 ,
=
Ki \ {P,P
C
= C3 ', m'
~
m + r3 - (q-2) 1
> (q-9)/2 + (q - "4 Iq - 5/4) - (q-2) =
(2q - Iq - 15)/4.
If R3 is a point of M', then R3 is the meet of two generators g and g' of an H3 in F.
Also g and g' meet C2 in distinct points R2 other than P and Pl ' and meet Cl in distinct points Rl other than P and P2• If g or g' contains P3 ' it is
456
J. W.P. Hirschfeld
coplanar with the tangents to C and C at P ; these tangents are distinct and l 2 3 their plane meets C in at most two points. If R3 is not of this type, then g and 3 g' each meet each C in some paint other than P,P ,P and P . There are at least i 3 l 2 m' - 2 such R3 each giving two different R2• each of which is obtained from at most two R3• Hence the number of such R2 is ;;. m' - 2
> (2q - Iq - 23)/4, which is positive for q ;;. 14.
Hence the number of R2 also in K ' is, again by 2
Lenma 2.4,
> (2q - Iq - 23)/4 + (4q - Iq - 1)/4 - (q-2)
= (q - Iq - 8)/2 , which is positive for q ;;. 13.
Each of these R2 lies on at least one line R2R3
meeting C2 and C3 in points of K2 and K3 , and meeting Cl in a point Rl • Each such Rl arises from at most two R2 • So the number of Rl in Kl' is, by Lemma 2.4, 1
> (q - Iq - 8)/4 + (q - '4 Iq - 1/4) - (q-2) = (q - 21q - 1) /4 ,
which is positive for q ;;. 7. Anyone of the corresponding lines R1R2R3 meets K ,K and K3 in distinct l 2 points.
0
LEMMA 2.8: Let K be a k-aap in PG(3,q), q odd and q
~
67 ,with
2 1 3/2 + R(q) k > q - '4 q R(q)
and
(31q + 14/q - 53)/16 2q -
::'et .~ be any line such that
I~
n KI
(I,q -
;,
7)2 + 41/16.
1 and 'let the q+ 1 planes through ~ be
i , i=l ••••• q+l. If Ki = TIi n K is a ki-arc, and, if ki > q i = 1, •.. , S, then S > 3(/q + 9)/4.
n
I~
PROOF: If k i
~
n KI = r, then r
= 1 or
As ki q - '4 /q + 7/4 for; = S+l, ••• ,q+l, so 1
2.
~
q + 1 for
t Iq + 7/4 only for 1, ... ,S and
Caps in ellip tic quadrics
457
,
k = r + E(k. - r) .;;; Ek. - q ,
1
.;;; S(q+l) + (q+l-S)(q-"4 Iq+7/4) - q = S(/q-3)/4 + q2 - 41 q3/2 + (7q - Iq + 7)/4. If the lemma is false and S .;;; 3(/q+9)/4, then 2
k .;;; q -
41 q3/2
= q 2 - 41 q 3/2 a contradiction.
+ (31q + 14/q - 53)/16 + R(q),
0
2 1 3/2 2 NOTES: (i) It is necessary to take q ;;;. 67 only so that q -"4 q + R(q) < q + l. It
only requires q ;;;. 11 for q - 41 Iq + 7/4 < q+l, and q ;;;. 9 for 3(/q+9)/4 < q + l. (ii) The number 3(/q+9)/4 is the smallest that is large enough for both the
next lemma and the final theorem. (iii) This is the lemma that determines R(q): it is the smallest value consistent with the required lower bound for S. 2
LEMMA 2.9: If K is a k-aap in PG(3,q), q odd and q ;;;. 67,with k > q -
41 q3/2+ R(q),
then there exist aonias Cl ,C and C3 forming a fZat tetrahedraZ system suah that,
2
,
,
1
if K. = K n C.is a k.-ara, then k. > q - -4 Iq + 7/4 for i = 1,2,3.
"
PROOF: As k> 1, take two distinct points P and P3 in K. By the previous lemma, there are S > ~(/q+9)/4 planes ni through PP 3 such that K n ni Ki is a ki-arc with ki > q - "4 Iq + 7/4. Take two such planes nl and n2; then Kl and K2 contain P and P3• Also Kl is contained in a conic Cl and K2 in a conic C2 , Theorem 1.1; hence K n Cl = Kl and K n C2 = K2 • Let Pl be a point of K2 other than P and P3 • Through PP l there are more than 3(/q+9)/4 planes n.' such that K n n.' = K.' is a k.'-arc with 1 ' " , ki ' > q - "4 Iq + 7/4. One of these planes, namely n2' passes through P3 and at most one is tangent to Cl at P. Therefore there are more than 3(/q+9)/4 - 2 =
= 3(/q+19)/4 planes ni ' each meeting Cl in a distinct point P2 other than P and P3" If there are m of these points P2 , then m > (3/q+19)/4. points P2 in K1 is, by Lemma 2.4,
So the number of
J. W.P. Hirschfeld
458 ~
m + kl - (q+l)
> (3/q+19)/4 + (q- ~/q + 7/4) - (q+l) =
(/q+ 11 ) /2.
Therefore there exists P2 in Kl and so the plane n3 = PP 1P is one of the 2 planes IT; '. Put K n n3 = K3 ; then K3 is obtained in a conic C3 • The three conics C1 ,C and C constitute a tetrahedral system. If the system were not flat, then 3 2 Kl u K2 u K3 is not a cap, Lemma 2.7. However, as Kl U K2 U K3 is contained in K, it is a cap and therefore the system is flat. 0 THEOREM 2.10: :n PG(3,q) with q odd and q ~ 67, if K is a k-cap with . .. K''l-S an e II"'l-pt'l-C k > q2 - '41 q3/2 + 2q, then t he on l y comp l ete cap conta'l-m.ng JuadT'ic Q.
In fact, it sUffices to take k
R(q)
2
1 3/2
> q - '4 q
+ R( q) ,
(3lq + 14/q - 53)/16 = 2q - [ (/q-7)
2
+ 4] /16.
PROOF: By Lemma 2.9, there exists a flat tetrahedral system of conics C ,C ,C l 2 3 1 such that K n Ci = K. is a ki-arc with k > q - '4 Iq + 7/4. By Lemma 2.6, the i 1 the three conics are contained in a unique quadric Q. show that K C Q; for, as k
~
It is now sufficient to
2
(q +q+4)/2, so, by Lemma 2.3, Q is elliptic and the
only complete cap containin9 K. Suppose there is some point Q in K \ Q.
As before, C ,C and C have the l 2 3 point P in common and have the residual intersections Pl ,P 2 and P • So Q is on no 3 C and Q F P,P ,P ,P • 1 2 3 i Let m be the number of planes n ' through PQ meeting C , apart from P, in a i l 1 point Q1 of K1 with Q1 distinct from P,P 2 ,P 3 and such that K n n i ' = Ki' is a 1 k.1 '-arc wi th k.' > q - -4 Iq + 7/4. 1 Let m be the number of these m1 planes ni ' meeting C2 in a point Q of K2 2 2 with Q2 distinct from P,P ,P . Similarly, let m3 be the number of these m planes 2 1 3 meeting C3 in a point Q of K3 with Q distinct from P,P ,P 2• 3 l 3 It will be shown that m3 > O. Firstly we note that IKl \ {P,P ,P 3 }1 > q - ~ Iq - 5/4. From Lemma 2.8 the number of planes n ' with i ki ' > q - '4 Iq + 7/4 is more than 3(/q+9)/4. So we obtain in succession the fol-
t
lowing, always using Lemma 2.4:
Caps in elliptic quadrics
459
1
m1 > (q- "4 Iq-S/4) + 3(1q+9)/4 - (q+l)
(/q+9)/2,
m2 > (q- ~ Iq-S/4) + (lq+9)/2 - (q+l)
(/q+9)/4,
1
m3> (q-"4 Iq-S/4) + (/q+9)/4 - (q+l)
O.
So there exists a plane nO', say, containing P and Q, and meeting Cl in Ql' C2 in Q2 and C3 in Q3 with each Qi distinct from P,P l ,P 2 ,P 3, such that KO' = K n nO' is contained in a conic CO'. The four conics CO' ,C l ,C 2 ,C form four tetrahedral systems. As 3 CO' U Cl U C2 C K, the system of CO' ,C l ,C is flat, Lemma 2.7. The conic C ' has 2 u at Q a tangent coplanar with the tangents at Q to C, and C?; this plane is the tangent plane to Qat P. Q3'
So CO' is tangent to Qat P and meets Q in Ql,Q2 and
Therefore CO' 1i es on Q. and, as Q is on CO', so Q is on Q. IJ
COROLLARY: In PG(3,q), q odd and q
~
67, if K is a complete k-cap which is not an
elliptic quadric, then
k .;;; q 2
_l
4
q3/2 + (31q + 14/q - 53)/16
_ ~ q3/2 + 2q. 0 NOTE: This theorem gives a better result than Theorem 1.8 for q
~
139.
3. THE MAXIMUM SIZE OF A CAP Three planes in PG(4,q) lie in a pencil if their join is n3 and their intersection TIl' LEI~MA
3.1: In PG(4,q), three quadric surfaces Ql,Q2,Q3 in respective solids
l , a2 , a3 have in common the conic C in the plane n. Then Q.l'~'~ lie on a quadric primal Q if and only if at each of three points of C the tangent planes to
a
the three
~
lie in a pencil.
PROOF: (i) Suppose Q exists. a simple point of Q. to
~
at P.
Consider a point P in C.
The tangent solid
a
Then Q n a i = ~, and P is a. in the tangent plane
to Qat P meets
,
These planes have the line ann in common and lie in the solid
a.
460
J. W.P. Hirschfeld
(ii) Here we give an analytic proof of the converse as opposed to the synthetic one in [4J.
~
Let
= V(x O2 - x1x2
The tangent plane at P(Y) to Tp(y)(~)
+
~
is
= V(2x oyo - x1Y2 - x2Y1 +
x3(a i yO + bi Y1 + c i y2 )
+
Y3(a i x + bi x1 + c i x2), x - di x3 )· O 4
(t,t 2 ,1,0,0) so that P(Y)
If Y
x (a i x + b x + c x2 ), x4 - d x )· 3 i 1 i i 3 O
E
C, then
where The T. have the line 1
in common. i
=
The
lie in a solid if there exist A ,A ,A and F such that, for 1 2 3
1,2,3,
This requires that
\.1 =
A
2 2 and a;t + bi t + c i - Ad; = G = F - AX 4 - (2txO - xl - t X2) ; that is, the vectors (A ,A ,A ), (d ,d ,d ), (1,1,1) are linearly dependent, where 1 2 3 1 2 3 2
Ai = ait + bit + c i • This occurs for three values and so for all values of t if and only if (a 1,a 2 ,a 3 ), (b 1,b 2,b ), (c 1 ,c ,c ) is each linearly dependent on (d ,d ,d ) and 3 2 3 1 2 3 (1,1,1 ). Now, Q. is a solid section of a quadric Q if and only if there exist 1
e,f,g,h such that, for i
1,2,3, 2
Q = V(x O - x1x2 +
As the d. are distinct, so h 1
+
x3 (aix O + bi x1 + ci x2)
(x - di x3 ) (exO + fX 1 + gX + hx 3 ))· 4 2
= O. Also
461
Caps in elliptic quadrics
a.1 - d.e 1
1:11
a
b.-d.f=b 1
1
c. - d.g 1
1
=C
But these relations mean precisely that (a ,a ,a ), (b ,b ,b ), (c l ,c ,c ) is 1 2 3 1 2 3 2 3 each linearly dependent on (d ,d 2 ,d 3 ) and (1,1,1). 0 1 LEMMA 3.2: In PG(4,q), q> 121 and odd, let K be a k-cap and
~
n K is an s-arc with s > q -
t Iq + 7/4.
TI
a plane such tlmt
Then there do not exist three distinct
IT such that K. = a. n K is a k.-cap with 1 2 3 2 1 3 /2. 1 21 1 3/2 11 ki > q - 4 q + R(q), 1 = 1,2, and k3 > q - 4 q + /I q + 1.
solids a ,a ,a containing
PROOF: Suppose that the theorem is false. contained in a unique elliptic
quadric~.
Then, by Theorem 2.10, each Ki is Now we consider the alternatives posed
by Lemma 3.1. (i) There exists a quadric Q meeting The set M= Kl
U
K2 U
i in ~, i = 1,2,3. K3 is an m-cap contained in K with a
m = s + I(ki-s) = k1 + k2 + k3 - 2s. As s
~
q + 1, so
2 1 3/2 2 1 3/2 11 m> 2(q - 4 q + R(q)) + (q - 4 q + /I q + 1) - 2(q+1) 3(q2 -
t
q3/2) +
i (37q + 14 Iq - 61).
There are two possibilities for Q.
(3.1 )
Either Q = P4' the nonsingu1ar
quadric, or Q = TIOE3' the singular quadric with vertex TIO and base E3 • (a)
Q= P 4
P comprises (q2+1)(q+1) points on the same number of lines with q+1 lines 4 through a pOint. Through each point of a line t on P4 there pass q other lines, whence q(q+1) lines t' on P meet t. No two of these lines t' meet off t, as 4 otherwise their plane would meet P4 in a cubic curve. Also P4 contains q2(q+l) points off t. So through each point of P4 ' t there is exactly one line t'. The m-cap M has at most two points on t and on each t', and every point of M lies on t
or some t'.
Hence
462
J. W.P. Hirschfeld
2
m ~ 2 + 2q(q+l) = 2(q +q+l).
(3.2)
From (3.1) and (3.2),
*
3(q2 -
q3/2) +
i
(37q + 14/q - 61) < m ~ 2(q2+q+l ).
Hence q2 _ l q3/2 + ~ (3q + 2/q - 11) < 0 , 4
8
a contradiction.
2
Through ITO there are q + 1 generators of Q, each containing at most two poi nts of
:11.
So 2 m ~ 2(q +1).
(3.3)
From (3.1) and (3.3), 3(q2 -
*
q3/2) +
i (37q+14/q-61)
q - 4 q +4 q
As
1 7 =q(q-4 /q + 4)'
there is some plane 1
7
k' > q - 4/q + 4'
TI'
other than
TI
meeting K3 in a k'-arc K' with
So, by Theorem 1.1, K' is contained in a conic C' = TI' n Q3'
Caps in ellip tic quadrics
463
It will now be shown that there exists a line P'P P ' where P' 1 2 and P2 E K2' Let V be the quadric of
% in
meeting
C + C',
E
K',P
l
E
Kl
Take a point P' in C' \ C,
Since it is simple, the tangent space Tp'(V) to Vat P' meets V in a cone P'P2' So there are q + 1 lines £' of V in Tp'(V),
As Tp'(V) does not contain C it meets
C in at most two points, whence at most two lines £' meet n,
The others, in num-
l in a point Pl of Ql and a 2 in a point P2 of Q2' with P ,P not in C, Also, P # P since a n a 2 = TI and P ,P ~ TI, Further, l l 2 l 2 l 2 P. # P', i = 1,2, since every point of a n C' lies on C, l 1 1 1 Let P' E K' \ C and note that IK' \ cl > q - 4 Iq - 4' For each such P', ber at least q - 1, all meet
a
there are at least q - 1 points P ' Conversely, each P is derived from at most l l two P', namely K' n Tp (V), unless Tp (V) contains TI' and hence K'. This exceptional case can
O~ly occur twice,lwhen Pl lies on the polar line of the plane
Thus each P' gives at least q - 3 points Pl ' apart from the exceptions; each P comes from at most two P', Thus, with A = {P in Ql obtainable from some P' in l l TI',
K' },
1 1 1 IAI > '2 (q - '4 Iq - '4) (q - 3)
= l q2 _ l q3/2 _ l (13q - 3/q - 3). 2
8
8
Let B = K , C = Q and Kl f' = {P in Kl obtainable from some l l l Kl ,', = A n B. So, by Lemma 2,4,
pI
in
K'}.
Then
IK/I >~q2 _ ~q3/2 _~ (13q - 3/q - 3) _ (q2 _ l q3/2 + R(q)) _ (q2+l) 4
= l q2 _ ~ q3/2 + ~ (5q + 20/q - 63) 2
8
16
'
The line P'P with P' in K' and P in Kl* meets a in a point P of Q \ C, l 2 l 2 2 Such a P2 is obtained at most twice when ITp (V) n K' 1 ~ 2, unless Tp (V) ~ TI',
which can occur for at most two points P ' w~ere the polar line of rr,2meets Q2' 2 Thus, with A = {P in Q obtainable from some P1P'}, 2 2 1A1
3 q3/2 + 32 1 (5q + 20 I q - 63), = '21 1K1f'l > '41 q2 - 16
464
p'
E
J. W.P. Hirschfeld
K', P,
E
K/'}, Lemma 2.4 gives that (5q + 20/q - 63)
+
1
=4
q
1 3/2
2
(q - 4 q
2
-
7
16
q
3/2
2
+ R(q)) - (q +1)
1 + 32 (67q + 48/q - 201)
> O. So there is a line meeting K',K 1,K in distinct points. 2 provides the desired contradiction. 0
So K is not a cap, which
NOTE: The condition that q > 121 is only necessary to ensure that both 2 1 3/2 2 1 3/2 11 2 q - '4 q + R(q) and q - 4 q + "4 q + 1 are less than q + 1. THEOREM 3.3: In PG(n,q), n
~
4, q> 121 and odd,
2(n,q) < q
m
n-1
-
41 qn-3/2
+
3q
n-2
•
In fact,
m (n,q) < qn-l ! qn-3/2 - 4 2
+
!4 (11q n-2
- 2(q
n-5
+
2qn-5/2
+ ••• +
-
l4qn-3)
1) + 1,
where there is no tem _2(qn-5 +... + l} for n = 4.
PROOF: Let K be a k-cap in PG(n,q). {il
(a) There is no plane
11
n
=4
such that
11
n K
Take a line t meeting K in two paints. through
l
1
7
is an s-arc with s > q - 4 /q + 4 . There are q2
each meeting K in an m-arc with m" q -
+
t /q + i.
q + 1 planes So
11
465
Caps In elliptic quadrics
k " 2 + (q
= q3
2
+ q + 1) (q -
1
4
Iq -
1
4)
1 5/2 1 2 3/2 - -4 q + - (3q - q + 3q - Iq - 1) 4
3 1 q5/2 3 2 < q-4 +q.
(b) There is a plane rr such that rr n K is an s-arc with s > q -
t i. Iq +
by Lemma 3.2 and Theorem 2.10, there are at most two solids through
an elliptic quadric, and, for the other q - 1 solids a through n.
la So k " 5 + 2(q
2
n
KI
"q
2
-
1 3/2
4q
+ 1 - s) + (q - 1) (q
2
11
+ "4 q + 1
-"41 q3/2
11 +"4 q + 1 - s)
2 1 5/2 1 3/2 1 7 JT1(y),m+n) for all X,jEH,m.nE{O,l, ... ,q-l}. On the other hand. every semidirect product with the above properties gives rise
On certain linear congruence class geometries
485
to a group of type IIIb. Type IIIc: If exactly three different prime numbers occur in G as orders of
elements. then G is an isomorphic copy of the alternating group A5 on 5 symbols. There is no finite group of type III with more than three different prime numbers occurring as orders of elements of the group. Type IV: Let G again be a (possibly cyclic) finite group. in which each non-
identity element is of prime order.
Consider. for some integer k ~ 2 (k
~
3. if G
is trivial). pairwise disjoint copies Gi of G. i = 1.2 •.••• k. Let G operate simultaneously as left translation group on each group Gi . The congruence class geometry of the algebra (·-1 2() k G1.• G) obtained in this way is linear. with G 1- , , •.• , as dilatation group. and consists of k sets of transitivity under G.
3. THE REPRESENTABILITY PROBLEr1 3.1. A geometry is said to satisfy the Exchange Axiom. if y E [XI and y Ej: [X \ {x}) always implies x E r (X \ {x}) u {y}). As every affine desarguesian geometry satisfies the Exchange Axiom. the following lemma is valid. LEMt4A 3.1.1: If a geometry is representaNe oVe:!' an affine desarguesian geometry, then it satisfies the Exchange Axiom.
PROOF: Let I.
(V.~)
be an affine desarguesian geometry with subspace closure operator
Let A be a subset of V. equipped with the induced subspace closure operator
I A and parallelism map
and x.y EA. X:: A such that y E [XI A and y 1 [X \ {xll A• [XI A g [XI nA and [X \ {x}I A = [X \ {xlI n A implies y E [XI and y 1 rx \ {x}). whence. by the Exchange Axiom in (V.~). x E [(X \ {x}) u {y}] n A = = [ (X \ {x}) u {y}] A•
~A'
o
By the Lemma. every representable finite congruence class geometry trivially occurs among those geometries of the list in chapter 2 which satisfy the Exchange Axiom in particular for sets X with
Ixi = 3. i.e. for all their subplanes. There-
fore in the rest of this section those geometries are selected from the list in chapter 2 which satisfy this condition. The geometries of type I trivially have this property. but the geometries of type II have not. as already mentioned in chapter 2.
If a geometry is of type
486
T.lhringer
IlIa. and the group G is abelian and of exponent P. then the congruence relations of the algebra (G.G) are identical to those of a vector space over GF(p) having G as additive group tcf. Huppert [9]. p. 80. or Wille (14). p. 36).
Therefore the
geometry ;s affine and desarguesian. and thus was already listed under type I.
If
G is not abelian. then the geometry contains a subplane which does not satisfy the Exchange Axiom: Consider two noncommuting elements x and y of G. and denote the identi ty of G bye.
Then the plane [e.x,y] is just the subgroup <x,y> of G
generated by x and y.
As<x.y>is a finite group of prime exponent. by Hauptsatz
2.3 of Huppert [9J it contains a central element z
ing x·z=z·x and y·z=z·y. z
~
[e.xJ.
Thus z
E
[e.x.YJ. z
~
~
e. i.e. an element z satisfy-
[e.xJ or z
~
[e.yJ. say w.l.o.g.
In contradiction to the Exchange Axiom for subplanes y 1! e.x.zl.
because [e.x.zJ is an abelian subgroup of G. namely the group generated by x and z. Now assume for the rest of this section that every geometry satisfies the Exchange Axiom for all its subplanes. If the geometry is of type IIIb. then. by the same argument as for type IlIa. the normal subgroup H of G is abelian.
If the identity of H is denoted bye.
then. for every x E H. (4)(x) .0) = (e.l) • (x.O) • (e.l)
-1
• Therefore the element
(}(x).O) is always contained in the subgroup generated by (x.O) and (e.l). hence in the subspace [(e.O).(x,O),(e.l)J. Exchange Axiom yields (1)(x).O)
E
As (e,l)
[(e.O),(x.O)].
~
[(e.O).(x.O).(4>(x).O)l. the Thus. for every x
E
H. 4>(x) is
contained in the subgroup of H generated by x. i.e. 4>(x) = xn for some n
E
{1.2 •.•.• p-l 1.
As
4>
is a fixed-point-free automorphism of order q. it is an
easy calculation that n does not depend on x. if x
e.
~
Thus. up to this point.
exactly those groups of type IIIb remain. where H is abelian. and 4> is of the form x - xn for some (fixed) n E {2.3 •..•• p-l} satisfying nq=l (mod p).
These pro-
perties are sufficient to yield the Exchange Axiom for all subplanes of the geometry. because. by the proof of the forthcoming Theorem 3.2.5. every such geometry is representable. The geometry of type IIIc contains a subplane which does not satisfy the Exchange Axiom: Consider the permutations x = (123). y = (12)(34). z = (13)(24) and the identity id on the set n.2.3,4.5}. x
~
Then y
E
[id.x.zJ and y 1 [id.z]. but
[id.y,zJ, contradicting the Exchange Axiom.
If the geometry is of type IV. then the subspace Gl of the geometry is isomorphic to the congruence class geometry of the algebra (G.G). All subplanes of this subspace satisfy the Exchange Axiom. and G thus is type IlIa or IIIb and has
On certain linear congruence class geometries
the properties described for these groups in this section.
487
These properties for G
are again sufficient to yield the Exchange Axiom for all subplanes of the geometry: The only nontrivial case of the Exchange Axiom for
IXI
~
3 is when
IXI = 3,
E G , xl ,x E G , i ~ j, Y E [x,x ,x ] and y E/: [xl ,x ], i.e. j l 2 2 2 l 2 i y E x·U., where U. is the copy in G. of the subgroup U. of G. generated by -1 1 .. 1 .1 J J Xl ·x 2• TrlV1ally then x E y.U , 1.e. x E [y,x l ,x ]. i 2 Thus the characterization of those finite congruence class geometries which
X = [x,x ,x ] , x
satisfy the Exchange Axiom for all their subplanes is complete. 3.2.
For each of the geometries described in section 3.1 either a represen-
tation is given (Theorem 3.2.5), or a representable geometry is found, having the same set of points and the same subspaces, but a different parallelism map (Theorem 3.2.6).
To this end some preliminaries are necessary.
The following lemma is essentially Theorem 11.2.3 in Artin [3]. LEMt·1A 3.2.1: In a representable geometry every dilatation is uniquely determined by the image of two distinct points.
PROOF: Let f be a dilatation of the representable geometry (A,rr), x,y,z E A, x
~
y and z E/:[x,y].
Such x,y,z exist by the General Assumption of n. 1.2.
f(x} and fey) are given, then fez)
E
If
rr(f(x}lx,z} n rr(f(y)ly,z), which is the
intersection of two subspaces, each of them being a point or a line, because the geometry is representable.
Thus fez) is uniquely determined, except in the case
that rr(f(x}lx,z} and rr(f(y)ly,z) are both lines and are equal. [x,z]
But then
= [y,z], whence y E [x,z], and thus z E [x,y] by the linearity of the
geometry, contradicting the choice of z.
If w is an arbitrary element of A, then
there are two elements in {x,y,z} which generate a line not containing w. fore, by the same argument as for fez), few) is uniquely determined. LEt4t·1A 3.2.2: Let V be a vector space of dimension;;' 2 over the field F.
There-
0
Then
every dilatation of the associated affine des argues ian geometry A(V) is of the form x .... rx+v for some
V E
V and rEF.
PROOF: Every map of the form x .... rx+v clearly is a dilatation of the geometry. Now let f be a dilatation, and y, z
E
V such that fey) E/: [y,z}.
If
488
f(z)
T. lhringer
E
TI(z!y,f(y», then the map 9 with x
~
rx+v, r=l and v=f(y)-y, satisfies
f(y)=g(y) and f(z)=g(z), thus by Lemma 3.2.1 f=g.
If f(z)
~
TI(ziy,f(y», then the
map 9 with x ~ rx+(l-r)c satisfies f(y)=g(y) and f(z)=g(z), if c
E
I y,f(y)l n I z,f(z)] and r is such that r(y-c)=f(y)-c.
3.2.1 f=g, and the proof is complete.
Therefore by Lemma
0
The next two lemmas are of central importance for the results of this chapter. LEt~r,lA
3.2.3: If V id a vector space, and (A,TI) is a geometry which is represent-
,_:':,Zd over A(V), tiler, a map f: A ~ A is a dilatation of (A,TI) iff it is the !'(:sL!'ictior, of a dnutation of A(V) to A.
PROOF: Because of Lemma 3.2.1 a dilatation of (A,n) is uniquely determined by the image of two distinct points y and z of A.
Using Lemma 3.2.2 one can easily prove
that there is a dilatation g on A(V) satisfying g(y)=f(y) and g(z)=f(z). proof of Lemma 3.2.1 then yields that f is the restriction of 9 to A. direction of the lemma is trivial.
The
The lIifll_
0
If f is a dilatation of the geometry (A,TI), and x E A such that f(x) # x, then the 1i ne [ x, f (x)] is ca 11 ed a trace of f. LEf.1I-1A 3.2.4: Ler; V Ie a vector space and f a dilatation of a geometry which is representable over A(V).
Then f is the restriction of a translation in V, i.e. of
a map cf r;he form x .... x+a, iff f has wo distinct parallel traces.
In this case
').;;: :.mces of f QJ:'e paralleL
PROOF: The assertion of the lemma is an immediate consequence of the and 3.2.3.
Lemma~
3.2.2
0
Now the main result of this paper can be proved. THEOREI1 3.2.5:
.4 finite congruence class geometry which satiSfies the Exchange
Axiol"! foT' every subpZane is representable over an affine desarguesian geometry iff ':tu dilar;aLion grouF is either cyclic or transitive on the points of the geometry.
PROOF: One has to check which geometries of the list in section 3.1 are representable.
On certain linear congruence class geometries
489
The theorem holds trivially for geometries of type I and type IlIa.
Let the
geometry be of type IIIb, and the group H have pm elements for some positive integer m.
Consider a vector space V of dimension m+l over GF(p), a vector sub-
space U ~ V of dimension m, and a vector
WE
V\ U.
H can then be regarded as the
vector subspace U, equipped with the vector addition as group multiplication. Since the group multiplication in H is written additively now, the group automorphism cp of H maps u to nu. Consider the subset A:= {u+n i wlu E U,i=O,1,2, ... ,q-l} of V, and identify the element (u,i) of H x F with the element u+niw
E
A.
The
congruence class geometry under consideration may now be denoted by (A,TI).
It is
sufficient to show that the geometry (A,TI) and the induced geometry (A,TI A) of A(V) on A are equal. The left translation (u,i) of H x F corresponds to the map f
.: A ~ A with v+njw ~ (niv+u)+ni+jw, which is the restriction of the dilatation
U,l
.
of A(V) mapping x to n1x+u.
Thus by Lemma 3.2.3 every dilatation of (A,TI) is a
dilatation of (A,TI A), whence, for every x E A, X ~ A, TI(xIX) ~ TIA(xIX), because (A,TI) is a congruence class geometry. If [ 1, [ 1A denotes the subspace closure operator of (A,TI), (A,TI A) respectively, then, by transitivity of the dilatations of each geometry, ITI(xIX) I = I[ xli, ITIA(xIX) I = I[ X1AI, and [Xl ~ [X1 A• Let {x l .x 2' ••• ,x t } be a minimal subset of A such that [xl.x2'···.x~ A = [Xl A• then the intersection of a subspace of rank t of A(V) with A, whence t-l t-2 . I[XlAI = p or q'P • For each 1 E {l,2, .... t-l}, xi+l Ej: [X l ,x 2' .... xf cause otherwise xi+l
E
[xl,x2, ... ,xil
~
[~A
'
is
be-
[xl,x2, ... ,xi1A' contradicting the choice
of {x l ,x 2 '· .. ,x t }· Therefore 0 J [xl1 J [x l , x21 J ... J [x l ,x 2, .. ·, xt 1. Every nonempty subspace of (A,TI) obviously has pr or q'pr elements for some nonnegative integer r. Using induction over t, it is now easy to prove that I[X l ,x 2, .•• ,X l I t :;;, t-2 t-l . ~ q.p or p ,whence [X l .x 2, ... ,xt l [xl,x2, ... ,xtlA only lf t-2 t-l . . I[X l ,x 2, ... ,x t ll = q'P and I[X l ,x 2, ... ,x t 1AI = p • ThlS would lmply qlp, contradicting nq=l (mod p): As n can be regarded as primitive q-th root of unity
J
in the field GF(p) of integers modulo p, qlp-l.
Thus TI(xIX) =TIA(xIX), which
proves the theorem for geometries of type IIIb.
(Cf. Theorem 5 in Pasini [111 to
this part of the prOOf). Now let the geometry be of type IV, and G be cyclic of prime order p. Consider a prime number q such that plq-l, a vector space V of dimension k over GF(q), a basis
el~e2,
••• ,ek of V, a primitive p-th root of unity r of GF(q) and
the subset A = {rJeili=1,2, ... ,k; j=O.1.2 ..... p-l} of V.
Then the geometry (A.TI A) is a representation of the congruence class geometry of the algebra
490
T.lhringer
('-1 2(1 1- "
••• ,
k G.,G): Choose an element g. of each set G., which generates G., 1
.
.
1
identify the element g.J with rJ e . 1
1
E
1
1
A, and the dilatation group G with the
dilatation subgroup of (A.1T ) generated by the restriction f to A of the A dilatation 9 of A(V) with g(x)=rx. Clearly, g maps A onto itself. and thus can be
,
restricted to A.
,
Therefore the congruence class geometry of ('--1 "2().. , k G.• G) may now denoted by (A,n). and for every x E A, X ~ A 1T(xIX) ~ 1T A(xIX). Let Ai for i=l,Z, ••• ,k be the subset of A corresponding to G.: A. = {rje. Ij=O,l,2, ••• ,p-l 1 1
1
1
For each X ~ A there are subsets Kl and KZ of {1,2 •••. ,k} such that IX n Ai I = 1 iff i E Kl and ix n \ I ;;. 2 iff i E KZ' Let x E Aj • If j E K \ (Kl U K2), and U is the vector subspace of V generated by {e Ii E Kl U K }, then i 2 TIA(xiX) ~ (x+U) n A = (xl, proving n(xIX) = nA(xIX). If j E Kl U K2 and KZ = 0, xi
E
X n Ai for
{x;-\,Ii,i'
E
Kl , and U is the vector subspace of V genera ted by Kl I, then 1T A(xIX) = (x+U) n A, x+U is of rank IK11 in A(V), and ever~1
i
E
contains one element of each Ai' i E Kl • Thus x+U cannot contain an element of A., i ~ Kl , or two different elements of one set A., because then x+U would be of 1 1 rank> IKll.
Therefore I11A(x!X)1 = !K l !. ,(xiX) = TIA(xIX).
Clearly, In(x!X)1 = IK1I, and thus
KZ and KZ f 0, and U is the vector subspace of V generated by lei Ii E Kl U K21, then x E U and nA(x!X) = UnA = i E K?U K Ai' which is Z trivially equal to ,(xix). If
j
E
Kl
U
Now let the geometry be of type IV, and G consist of one element.
Then the
geometry is discrete, i.e. n(X!X) = {xl and [Xl = X for every element x and subset X of . --1 1
"
z()
•• ,
kG .• and it can be represented for example on a bas i s of the vector 1
space of dimension k over an arbitary finite field, as is easily proved. Let the geometry be of type IV and the dilatation group be noncyclic and of type IlIa.
G then is elementary abelian and contains at least p2 elements.
Thus
there are elements x,y of G and g E G such that g·x ~ [x,yl (recall that each Gi l is an isomorphic copy of G on which the elements of G operate as left translation~. Then goy E n(ylx,g'x), because y=(y.x
-1
).x, g.y=(y.x
to be represented over A(V) for some vector space V.
-1
)·g·x.
Assume the geometry
By Lemma 3.2.4 g is then the
restriction of a translation in A(V), and thUS, by the same lemma, g,z2 E n(zZlzl,g'Zl) for elements zl E Gl ,z2 E G2• This contradicts 11(z2Iz1,g-Zl) = (z2 1• If the group is of type Illb, and the geometry is assumed to be represented over A(V) fora vector space V, then the element (x,D) E H
x
F, for some xE H\ {el,
491
On certain linear congruence class geometries
has two parallel traces: (x.l)=(x,O)·(e,l)
E
1T((e.l)l(e.O).(x.O)·(e.O)). if all
elements occuring are again simultaneously regarded as elements of G and Gl . This leads to a contradiction by the same argument as was used for groups of type IlIa.
0
THEOREM 3.2.6: For every finite congruence cZass geometry satisfying the Exchange Axiom for each subpZane, there exists a geometry having the same points and subspaces which is representabZe over an affine desarguesian geometry.
PROOF: This theorem is a consequence of Theorem 3.2.5. except for geometries of type IV. having a noncyclic dilatation group.
Consider such a geometry equipped
with a dilatation group G of type IlIa. such that G has pm elements for some integer m ~ 2. and a vector space V of dimension k+m over GF(p). a basis e l .e 2•.•. ,e k+m of V. the vector subspace U of V generated by {ek+l.ek+2 •...• ek+m}. and the subset A 11:1 1.-1 , 2U k (e.+U) of V. The group G can be regarded as the , ••• , , additive group on U. and an element of Gi • which is a copy of x E G. is identified with the element e.+x E A. The congruence class geometry of the algebra 1
(.--1 2U k G.• G) can thus be regarded as a geometry (A.1T) on A. and. by similar 1 " ••• , 1 arguments as were used in the proof of Theorem 3.2.5. it can be shown that the subspace closure operators [ I of (A.1T) and [ I A of the induced geometry (A.1T A) of A(V) are equal: Every dilatation of (A.1T) is a dilatation of (A.1T A). therefore [XI
~
I[XII
[XI A for every set X ~ A. 11:1
By elementary considerations one obtains
I[ XIAI, thus proving [XI = [XI A• The details of this part of the proof
are omitted, because they are easy but rather technical. Now let the geometry be of type IV and the dilatation group G of type IIIb such that the group H has pm elements for some positive integer m.
Consider a
vector space V of dimension k+m over GF(p). a basis e l .e 2•...• e k+m of V. the vector subspace U ~f V generated by {ek+l.ek+2 •..•• ek+m} and the subset k (nJe.+U). A = 1.-1 , 2U , ••• , 1
The group H can then be regarded as the additive group
on U. and an element of G.1 which is a copy of (x.j) . J
the element n ei+x.
E
G = H x F is identified with
The congruence class geometry of the algebra (·=1 2U 1
"
••• ,
kG .• G)
can thus be regarded as a geometry (A.1T) on A. and again. by the methods used above. it can be proved that (A.1T) and the induced geometry (A.1T A) of A(V) have the same subspaces. [] The following corollary could. of course. have been proved directly.
,
T.lhringer
492
COROLLARY 3.2.7: A f'inite congl'uence class geometl'Y satisfies the Exchange Axiom iff
aLZ subpZanes 0;' the geometl'Y do.
PROOF: Immediate from Theorem 3.2.6, Lemma 3.1.1 and the fact that the Exchange Axiom only involves subspaces, but not the parallelism.
0
BIBLIOGRAPHY 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
J. Andre, Uber Parallelstrukturen. Tei1 I: Grundbegriffe, Math. z., 76 (1961), 85-102. J. Andre, Uber Para11e1strukturen. Tei1 II: Translationsstrukturen, Math. Z., 76 (1961), 155-163. E. Artin, Geometric Algebm, Interscience Publishers, Inc., New York, 1957. 1·1. Bil i otti, Strutture di Andre ed S-Spazi con tras 1azi oni. Geom. Dedic. (to appear). P. Dembowski, Finite Geometl'ies. Springer-Verlag, Ber1in-Heide1berg-New York, 1968. G. Gratzer, Univel'sal AZgebl'a. D. van Nostrand, Princeton, N.J., 1968. M. Hall, Jr., The Theory of Gl'OUps. MacMillan, New York, tenth printing, 1968. A Herzer, End1iche nichtkommutative Gruppen mit Partition TI und fixpunktfreien 11 -Automorphi smus, Arch. Math., 34 (1980), 385-392. B. Huppert, Endliche Gruppen I. Springer-Verlag, Ber1in-Heide1berg-New York, 1967. P. Libois, Quelques espaces 1ineaires, Bull. Soc. Math. BeZg., 16 (1964), 13-22. A. Pasini, On the Finite Transitive Incidence Algebras, Boll. Un. Mat. Ital., (5) 17-B (1980), 373-389. G. Pickert, Pioojektive Ebenen. Springer-Verlag, Ber1in-Gottingen-Heide1berg, 1955. H. Werner, Produkte von Kongruenzk1assengeometrien universel1er A1gebren, Math. ;:-., 121 (1971), 111-140. R. Wille, Kongl'uenzklassengeometY'ien. Lecture Notes in l-lathematics 113, Springer-Verlag, Berlin-Heidelberg-New York, 1970.
Technische Hochschule FB Mathematik, AG 1 Schloagartenstr. 7 0-6100 Darmstadt Federal Republic of Germany
Annals of Discrete Mathematics 18 (1983) 493-510 © North-Holland Publishing Company
493
TRANSLATION PLANES OF CHARACTERISTIC P THAT ADMIT SL(2,pr) N.L. Johnson
1. INTRODUCTION One of the most basic problems in the study of translation planes (characteristic p) is to determine the possible collineation groups generated by collineations of order p. In this article, we explore the situations where the p-elements generate SL(2,pr) and conversely investigate the permutation group action of a collineation group isomorphic to SL(2,pr) acting on the points and lines of a translation plane.
We are also concerned with the combinatorial nature of the spaces fixed
pointwise by the Sylow p-subgroups. 2. THE KNOWN TRANSLATION PLANES ADMITTING SL(2,pr) The Desarguesian plane terized by such a group.
~
of order n admits SL(2,n) and conversely is charac-
One of the first classification results is:
2.1 LUneburg [32]: Let E denote a projective plane of order pe which admits a collineation group isomorphic to SL(2,pe). ~
is determined.
Then ~ is Desarguesian and the orbit structure of
In particular, if the group acts on an affine plane then the p-
elements are elations. The question of whether affine elations generate SL(2,pr) (in a translation plane) has been studied by Ostrom in [33) , [34) and by Hering [15) . 2.2 Hering-Ostrom: Let
~
be a translation plane of characteristic p and let E denote the
collineation group in the-translation complement generated by the affine elations of
~.
Then E is either
494
N.L. Johnson
(1) elementary abelian,
(2) SL(2,pr), (3) SL(Z,5) and p = 3, r
(4) SZ(Z ) and p = Z, or
(5) E contains a normal subgroup N of odd order and index 2. By Hering [161, if the group E is Sz(Zr) there is an invariant LUneburg-Tits Zr subplane of order Z If the group E is SL(Z,pr) there is an invariant Desarguesian subplane of order pr by Ostrom 1331 . Foulser [71 has shown that for p > Z the results of Hering-Ostrom remain valid for collineation groups generated by Baer p-elements (the group element fixes a Baer subplane pointwise) .
.?-)
Foulser ( 71:
Let
TI
be a translation plane of characteristic p.
Let B denote the collin-
eation group in the translation complement generated by all Baer p-elements.
If
p > 3 then the Baer subplanes pointwise fixed by p-elements are disjoint (as subspaces).
Furthermore, B is either elementary abelian or SL(Z,pr).
Moreover, the
Baer subplanes pOintwise fixed share the same infinite points. 2.4 The Hall Plane (of order q2 admits SL(2,q) where the p-elements (pr=q) are Baer): Z
The Desarguesian plane n of order q admits SL(2,q) and the Sylow 2-subgroups define a derivable net N in n.
The p-elements in n act as elation groups. Z
Deriving N produces the Hall plane of order q where the p-elements now fix Baer subplanes pointwise . .Z.5 The Hering Planes (( 13J , [ 141) (order q2 and admit SL(2,q)): Let P = PG(3,q), q odd> 3, and let C denote a twisted cubic in P (set of qtl points in P such that no four points are coplanar, Segre [38]). tangents T to C forms a partial spread.
Now further assume that q =
The set of mod 3 and
let S denote the subgroup isomorphic to SL(Z,q) of GL(V ) which leaves the 4 twisted cubic invariant. Let g be an element of order 3 in S acting on P.
Then
g fixes exactly two lines of P (Z-subspaces of V ) and fixes one of them pointwise. 4 1 In SL(2,q) there are zq(q-l) subgroups of order 3 and each such subgroup fixes
Translation planes that admit SL(2,pr)
two distinct lines in P.
495
The set of q+l tangents to C together with these two
sets of ~(q-l) lines form a spread (see e.g. LUneburg [31) (45.1)).
Moreover, S
leaves invariant precisely one spread of P and acts irreducibly on V4. 2.6 The Ott-Schaeffer Pl anes ([ 37] , [39)) (order q2 and admit SL (2 ,q) ): Bya theorem of Segre [38) any twisted cubic C in P may be brought into the canonical form 3
2
2 3
«s ,s t,st ,t ) is,t
E
GF(q), (s,t) f (0,0»
for q odd> 3. By using a similar curve C(a)
=<ssa,tsa,sta,tt~ for q even where a is an
automorphism which generates the automorphism group of GF(q), planes similar to the Hering planes may be constructed.
Let S2 be a Sylow 2-subgroup of the group S
isomorphic to SL(2,q) in GL(V 4 ) fixing C(a). Then S2 fixes a point P of C(a) and fixes each line of a plane in P incident with P. Moreover, the elements of S2 flare elations with axis thru P.
The two remaining lines incident with Pare
in distinct S orbits and each orbit defines a regulus in P (derivable net in V4). The trick of the 3-elements noted in the discussion of the Hering planes gives the -1
It turns out that a,a give rise to the same curve C(a) so there are ~(r) if q = 2r such curves and two remaining q(q-l) lines of a spread for either regulus.
planes (one the derived plane of the other) for each curve. Ott-Schaeffer planes. automorphism).
Thus, there are
~(r)
(The Ott planes [37) actually correspond to the Frobenius
Here, as in the Hering case, the group S
~
SL(2,q) acts irreduc-
ibly on V4 . The Ott-Schaeffer planes may be characterized in the following way: THEOREM 2.7 (Johnson [19]): Let
n
be a translation plane of even order 22r that
admits a collineation group S in the translation complement isomorphic to
r SL(2,2 ).
Assume the Sylow 2-subgroups of S fix Baer sub lines pointwise (2
fine points on a line of n).
Then n is an Ott-Scaheffer plane.
The proof of (2.7) relies on the group orbit structure established by Johnson and Os trom in [27] .
r af-
496
N.L. Johnson
THEOREM 2.8 (Johnson-Ostrom): Under> the hypothesis of (2.?J the points at infinity of
1
1!
1
aT'e hi exact Zy ;;hr>ee orbi ts of r>espective Zengths, 1+q, ~ (q- 1), ~ (q-l ) .
Once one knows where to look, it is not hard to show that the orbit of length q+l defines a derivable net V in
1!
(note that we are not assuming anything
about the dimension of the vector space). V is derivable and S leaves V invariant so by Foulser we can imbed S in
GL(4,q). Now we establish that S must act irreducibly on V . This tells us (by 4 Brauer and Nesbitt [21)) the form S can take on V and using this we can work out 4 a form for components invariant under certain key collineations. The components are then seen to be subs paces over V which implies 4 (see section 3).
1!
is an Ott-Schaeffer plane
2.9 The Walker Planes of Order 25 [41) (admits SL(2,5»: 2
Let" be a translation plane of order q and kern
~
GF(q) that admitsSL(2,q)
Assume S acts reducibly but indecomposably on V4 . 5 (Walker [41]) and there is a canonical form for S (see e.g. LUneburg
in its translation complement. Then
q =
S may be represented as generated by
[31] 46.12, p. 240). 0 s 3s s
0
2 3
s 3s
2
s
0 -1 lO '"' c· :
([ '-'y s
Let w,
~l
0
as ~ . normalizes
0
0
0
0
0
-1
0-1
:J
-1
s
~,p
to be a component of
Then
~
fixes exactly six components of
permutes these six components. 1!
so that
p
fixes (X
1!
and
The form for S forces X = 0
= 0) and permutes the remaining five.
It turns out there is exactly one translation plane of each of the three cases: fixes 1, 3 or 5 of these components.
p
One of these three planes turns out also to
admit a group acting irreducibly on V which forces it to be the Hering plane of 4 order 25.
Translation planes that admit SL(2.pr)
497
2.10 The Dempw01ff Plane of Order 16 {3] (admits SL(2,4)): Let n be a translation plane of order 16 and let X = 0, Y = 0, y = x, y = xT be the components where T is a fixed point free element of GF(4,2). GL(4,2) ~ A8 and using ~ one can determine the conjugate forms for fixed point free elements in GL(4,2).
To construct a translation plane of order 16 one needs l a set Mof 15 such elements such that if a,S E M then as- is fixed point free. In {3] Dempwolff determines (originally with the aid of a computer program) a number of such sets M.
One of these sets is left invariant by GL(2,4) which
implies GL(2,4) acts as a collineation group of the corresponding plane fixing X = 0 and Y =
o.
This plane may actually be obtained by derivation of the semifield plane of order 16 with kern GF(2) (see Johnson [22]). .
4
2.11 The Foulser-Ostrom Planes of Order u (admitting SL(2,v), GL(v)
~
GL(u)):
There is a nice way of obtaining these planes using PlUcker coordinates, but the following description is more succinct. Let M(q) denote the Miquelian-Mobius plane of odd order q.
Then if P,Q are
pOints of a circle C there is a unique circle D F C orthogonal to C such that Dnc={p,Q}. Let Pq = PG(1,q) (C - PG(1,q2) - = M(q)). Let A = {circles N E M(q)/N is orthogonal to P and N n P = {Q,Q Vq} for Q E P - PI.-(P r = PG(l,..,.iq))}. Then the q q q vqyq circles of A are disjoint and permuted transitively by PGL(2,v'q) (Ostrom l36] ). The circles correspond to derivable nets in V4(q2) and by replacing each of these derivable nets by its unique replacement net, the constructed plane of order q2 will admit SL(2,)q) as a collineation group where the p-elements (pe=q) are elations. Let B = {circles N E M(q)/ N is orthogonal to Pq and N n P = {S,T}, SF T in PG(l,y'q)}. Clearly, PGL(2,jq} permutes the circles of B. q
By selecting sets of disjoint circles in AU B which are orbits under PSL(2,v) for GF(v)
~
GF(vq), various translation planes can be constructed admitting SL(2,v) as
a collineation group where the p-elements are elations (see Foulser [81]). For example, planes of order u8 admitting SL(2,u) may be constructed which may not be obtained simply by replacing a set of disjoint derivable nets corresponding to circles in A.
498
N.L. Johnson
2 Similarly for q even, planes of order u4 admitting SL(2,v) for v f u, GF(v)
~
GF(u) may be constructed.
Note in all of these planes the order is u4 , the group is SL(2,v) for some GF(v)
~
GF(u), and the kern
~
2
GF(u ).
Now let nO denote the plane obtained by replacing all the nets corresponding to circles in A (so has order u4 , admits SL(2,u), has kern GF(u 2 ) and there are
u(u;~ replaced derivable nets). There is in "0 a derivable D that shares u+l infinite points with PG(1,u 2 ) and two infinite points with each of the U(~-l) replaced derivable A-nets.
Actually these points are conjugate under x-->x
that GL(2,u) leaves D invariant.
u
so 4
By replacing D, a translation plane of order u
admitting GL(2.u) may be constructed where the p-elements (u
= pi) in the group
are Saer collineations and the kern is GF(u) as it turns out the Baer subplanes of Dare GF(u) but
no~
2
GF(u )-subspaces (see Foulser [9)).
2r 2.12 The Johnson-Ostrom Planes of Order 3 Admitting SL(2,3) (Bruck type {l)
planes):
These planes are of particular interest as apparently they are the only known translation planes that admit both affine homologies and elations with the same center.
The planes were obtained by trying to derive generalized Andre planes to produce planes with affine central collineations. THEOREM 2.13 (Johnson-Ostrom [28]): Let
n
be a Desarguesian affine plane coordi-
'uzr-;:zeG by GF(q2), q ;;. 25 and let; be a generalized Andre plane constructed by r'cplacement of disjoint nets Nl ,N , ... ,N in " such that the slopes in all of 2 k til£se are non-sqUal'ec. :'et NO be the net in ,,(or ;) denoted by {mim E GF(q) L /yl FE L} is a
spread of the 4m-2 dimensional space yl/y by totally isotropic (2m-l)-spaces (constructed by sections of the quadric by hyperplanes). So, this produces a translation plane of order q
2m-l
with kernel
~
GF(q).
In [29) Kantor defines several new n+(4m,q) spaces and finds spreads for certain of them. GU(2,q)
One of the spreads for n+{8,q) is found using
SL(2,q) X Zq+ 1 (see [29), section 6). By the choice of the "slice" y, SL(2,q) can act on the corresponding translation plane of order q3. The construc:=
tion of n+(8,q) requires q
=2 mod
3, so q
= 22e+l.
2.15 The Dye-Kantor Plane of Order 83 that Admits SL(2,8): Dye [5), section 4, constructs a spread L of a n+(8,8) space which is preserved by the group A9 . A particular slice of L is left invariant under SL(2,8) (~A9). The Dye and Kantor planes of order 8 are nonisomorphic (see Kantor [29), section 9}.
500
N.L. Johnson
2.16 The Liebler Planes (even order q3 admitting SL(2,q)): The Kantor planes mentioned above of order q3 have a S(=6L(2,q))-invariant Desarguesian subplane TIO'
Furthermore, there is a collineation group of order
q+l fixing TIO pOintwise and there are two S-orbits on too of, respectively, lengths 3
q+l and q -q. Liebler [30) uses his theory of enveloping algebras to construct planes of even order q3 possessing the above properties of the Kantor planes.
It is not
entirely clear whether the set of Liebler's planes contains those of Kantor. Liebler assumes that S is a partial spread associated with a subplane of order q and assumes the enveloping algebra £ K ~ GF(q).
Further, he assumes H ~
= £(S)
Mat ,2(K) $ Matz,z(K), 2 GL(2,q) acts on V so that the Sylow 2-sub~
groups determine S (of degree q+l) and H/Z(H) acts regularly on a set R of 3 spaces.
V = X s Y where X is the irreducible £-module consisting of points of
the subplane of order q and Y is a sum of two isomorphic irreducible £-modules. There are q+l isomorphic irreducible £-modules in Y which define a regulus M.
Q denote the locus of points on lines joining X to M (points in PG(V)).
Let
Then Q is
a degenerate quadric with vertex X. Liebler uses the quadric Q to give necessary and sufficient conditions that
SUR is an H-invariant spread and then exhibits a 3-space R satisfying the condiThe SL(Z,q)-module has the form Va $ V $ V where V 2 z z 2 a+l is the canonical representation. The condition on a F 1 is that X = 1 has no
tions (see (3.2) [30) ).
nontrivial solution in K ~ GF(q).
3. THE TRANSLATION PLANES WITH KERN
j
GF(q) OF ORDER qZ ADMITTING SL(Z,q)
Walker [41) and Schaeffer [31] have completely determined the translation planes of order q2 that admit SL(Z,q) as a collineation group in the translation complement and that have GF(q) in the kernel. 3.1 Walker [39], Schaeffer [31): Z
A translation plane TI of order q and kern
~
GF(q) admits S
~
SL(Z,q) iff TI
is Desarguesian, Hall, Hering, Ott-Schaeffer or one of the two Walker planes of
Translation planes that admit Sa2,pr)
501
25. TI is Desarguesian if and only if S is completely reducible and the p-element are elations. TI is Hall if and only if S is completely reducible and the p-elements are Baer. TI is Hering or Ott-Schaeffer if and only if S is irreducible. TI is Walker or Hering of order 25 if and only if S is reducible but not completely reducible. 4. TRANSLATION PLANES OF CHARACTERISTIC TWO IN WHICH ALL INVOLUTIONS ARE BAER In [25] Johnson and Ostrom study translation planes as in the title, adapting some of Schaeffer's arguments. 4.1 Johnson and Ostrom [25] : Let TI be a translation plane of dimension 2 over GF(q) (kern q = 2r.
~
GF(q», where
Assume S is a nonsolvable subgroup of the translation complement of TI and
contains only Baer involutions.
Let Sl denote the normal subgroup generated by s the Baer involutions. Then Sl is isomorphic to SL(2,2 ) and sir. 2s If Sl is irreducible, TI has an Ott-Schaeffer subplane TIO of order 2 and s
Moreover, for every factor So of s F 1, TI contains Ott-Schaeffer subplanes of order 22s 0 The elements of order 3 as affine homologies and determine is odd.
the component of TIO not fixed by a Sylow 2-subgroup. If Sl is reducible then TI is derived from a plane admitting SL(2,2 s ) where the involutions are elations. In the Foulser (-Ostrom) plane of order q4 admitting SL(2,q) where the pelements are Baer there are invariant Hall subplanes upon which SL(2,q) acts faithfully.
Apart from this example, I know of no plane TI that contains a non-
Desarguesian SL(2,pr)-invariant subplane TIO such that the planes TIO and TI are in different isomorphism classes. For example, if TI is a translation plane of order 22r admitting SL(2,2 s ), sir where SL(2,2 s ) leaves invariant an Ott-Schaeffer subplane TI O' is TI also Ott-Schaeffer? In [17] , Hering and Ho consider translation planes of dimension 2r, r odd (vector space has dimension 4r).
N.L. Johnson
502
4.2 Hering-Ho (5.7) [17]: Let
n
be a translation plane of even order and dim 2r, r odd.
Let S be a
co11ineation group in the translation complement containing only Baer involutions. Let S denote the subgroup generated by the involutions in S.
Let S be non-
solvable, then either (a) 5/0(S) or
=PSL(2,q),
=
q ±3 mod 8 s (b) S ~ 5L(2,2 ) for some s > 1.
Hering-Ho [17l and Johnson-Ostrom [ 26}a1so classify the co11ineation groups in even order planes when the involutions are not necessarily Baer. Oempwo1ff [4l studies translation planes of even order and arbitrary dimension that admit Baer groups of large even order. ~.3
Oempwo1ff [4]:
Let
n
be a translation plane of order 2n and S a co1lineation group in the Let E be a subgroup fixing a Baer subp1ane pointwise and
translation complement.
1et S'" denote the group generated by all subgroups that fi x a Baer subp 1ane poi nt-
wise and have order and IE; > 2n/4. (1) n
or
(2)
s*
lEI. Assume
Then one of the following possibilities occurs:
4 and S
~
S contains only Baer involutions, is nonsolvab1e
1:.
=SL(3,2)
5L(2,2 s ).
=
Note the Lorimer-Rahilly or Johnson-Walker (see e.g. [24]) are planes satisfying condition (l). Also, Oempwo1ff's results more generally include the case where S may contain elations. Although Ostrom's work [35J does not fit this section, the ideas are similar to those involved in the proof of (4.1). 4.4 Ostrom [35J (2.8): Let p > 5.
r
be a translation plane of order q2 whose kern contains GF(q), q
~
pr ,
Suppose that the linear translation complement has a subgroup S iso-
morphic to SL(2,ps).
Then either
(1) the p-e1ements are elations and S has an invariant Oesarguesian subplane of order pS (2) the p-e1ements are Baer and
n
is derived from a plane in which S acts as
503
Translation planes that admit SM2,pr)
in (1), or (3) S
2s has an invariant Hering plane of order p
The action of S on the
vector space is irreducible. The cases p = 3 or 5 are probably not much different (at least for SL(2,3 s ), s > 1) but the proof of (4.4) uses Suprunenko and Zalesski 's classification (see [40)), valid only for p > 5.
5. A CHARACTERIZATION OF THE DESARGUESIAN PLANE 5.1 Foulser-Johnson-Ostrom [12) : Let
IT
2
be a translation plane of order q , q f 3,4, that admits SL(2,q) as a
collineation group in its translation complement. (1) If the p-elements (q
= prj are elations then
IT
is Desarguesian.
(2) If the Sylow p-subgroups fix Baer subplanes pointwise then
IT
is Hall.
PROOF: (1) Sketch for q f 8. There is a Desarguesian net N of degree q+l (Ostrom [33)).
Let e be a
collineation of SL(2,q) such that lei is a prime 2-primitive divisor of q2_ 1. Then e fixes exactly two components of IT-N, acts irreducibly on each fixed component and generates over GF(p) a field isomorphic to GF(q2).
This may be done in
such a way that the (O,i )-homologies in the corresponding Desarguesian plane fix N componentwise and ;iX the two e-invariant components.
There are
~(q-l)
k
groups of SL(2,q) conjugate to <e> and to each corresponds a distinct pair of components.
Each field generated by a given 2-primitive element contains the same
field K of order q.
That is, K fixes each component so the result follows from
Walker-Schaeffer. The proof of (2) requires that the Baer subplanes pointwise fixed by Sylow p-subgroups belong to the same derivable net.
This is true by Foulser [7 ) (see
2.3)) if q f 3 and p f 2 and by Johnson [21) for p = 2, q f 2.
6. THE TRANSLATION PLANES OF ORDER q2 AND CHARACTERISTIC TWO ADMITTING GL(2,q) As seen in section 3, Walker and Schaeffer have determined the planes of the title with kern
~
GF(q).
However, without making an assumption on the size of the
N.L. Johnson
504
2
kern, the same results may be obtained for planes of even order q , q
~
4 with the
additional assumption that GL(2,q) is a co11ineation group (instead of merely SL( 2 ,q)).
By using the action of the center of GL(2,q), detailed combinational arguments yield a contradiction unless the Sylow 2-subgroup fixed point spaces are Baer sub1ines, lines or Baer subplanes.
By (2.7) and (5.1), we obtain:
6.1 Johnson [181: Let
IT
2r r be a translation of order 2 1 16 admitting GL(2,2 ) in the transla-
tion complement.
Then
IT
is Desarguesian, Hall or Ott-Schaeffer, depending on
whether the Sylow 2-subgroup fixed point spaces are lines, Baer subp1anes, or Baer sublines.
7. THE TRANSLATION PLANES OF ORDER 16 ADMITTING NONSOLVABLE GROUPS As seen in sections 5 and 6, planes of order 16 present an isolated problem. Are translation planes of order 16 that admit SL(2,4) or GL(2,4) Desarguesian or Hall?
(There are no Ott-Schaeffer planes of order 16). We mentioned the Dempwolff plane (2.10) admits GL(2,4) so we have at least
to add another plane to the list. The Lorimer-Rahilly and Johnson-Walker planes (see (241) of order 16 admit PSL(2,7) but the list stops here. 7.1 Johnson [24] : Let
IT
be a translation plane of order 16 admitting a nonso1vable col1in-
eation group.
Then
r
is Desarguesian, Hall, Dempwo1ff, Lorimer-Rahilly or
Johnson-Walker.
8. THE TRANSLATION PLANES OF ORDER
l
THAT AD/HT SL(2,q) AS A COLLINEATION
GROUP Fou1ser and Johnson [ 101 , [11] have detennined the planes of the title. Apart from the planes with kern
~
GF(q), only the Dempwolff plane is obtained.
We understand that Hering and Schaeffer have also worked on this general problem.
Translation planes that admit SU2,pr;
505
8.1 Foulser-Johnson [10] ,[ 11]: Let
IT
be a translation plane of order q2 (where pr
= q), which admits a
collineation group S isomorphic to SL(2,q) or PSL(2,q). Then IT is Desarguesian Hall or the Dempwolff plane of order 16 if and only if S is completely reducible.
S
IT
is Hering or Ott-Schaeffer if and only if S is irreducible.
IT
has order 25 and is Hering or one of the two Walker planes if and only if
is reducible but not completely reducible.
PROOF: Sketch (for q F 4) Let F
= GF(p), E = GF(pr). The irreducible representations of S ~ SL(2,pr)
have been determined by Brauer and Nesbitt [2]. Let N be an irreducible EG-module of dimension d and let K be the smallest field over which the matrices of S may be written.
Form the restriction module
NK. Assume K ~ GF(ps) and replace the elements of K in S by s X s matrices over F and replace the elements of K in a d-tuple in NK by an s-tuple over F. This makes NK into an irreducible FS-module of dimension ds.
Conversely, any irreduciple
FS-module may be obtained in this way (see Foulser-Johnson [10] (1.9)). Fong and Seitz [6] (usi ng Brauer and Nesbitt) have determi ned the i rreducible FS-modules of degree d
~
4r (= dimension of
IT
over G).
There are seven pos-
sible irreducible cases. It is shown that the only irreducible cases that support translation planes IT
are those involved in the dimension 2 situation and then shown that the plane
must also have dimension 2 in this case.
That is,
IT
is Hering or Ott-Schaeffer.
For the completely reducible cases, we may use the results of Fong and Seitz to determine the action on the invariant subspaces. structure to show the modules must be canonical.
We then use the group
This shows the plane
IT
to be
Desarguesian or Hall. Let S be reducible but indecomposable on submodule and
IT/W
Case 1: p
l
IT.
Let Wl be an irreducible FS-
= W2.
= 2.
Alperin's theorem [1] for p
= 2 on extension spaces shows that Wl ~ W2.
Dimension arguments then yield a contradiction, so the situation cannot occur. Case 2: p
F 2.
N.L. Johnson
506
=
Here W W and the possibilities for Wi given by Fong-Seitz. The module l 2 structure imposed yields impossible situations unless both Wl and W2 act canonically.
In this case, we really have a situation as in the dimension 2 case where
matrix arguments may be used (as Walker did in [41]) to show the characteristic is 5 and then that the plane have order 25. (In many of these arguments, we show the plane has dimension 2 and then use Walker-Schaeffer). Note this work reproves (6.1).
However, (6.1) may be obtained without
using representation theory.
9. PROBLEMS We have described the Kantor, Dye-Kantor and Liebler planes of even order q3 that admit SL(2,q) (possibly the Dye-Kantor and Kantor planes are also described by Liebler).
Are there others (besides the Desarguesian)?
Schaeffer [39] has described the possible modules on planes of order q3 with kern
~
GF(q) but not the planes in this situation.
In the even order case the
module is v~ ~ v~ w v~ where V is the canonical representation and a,B,o are the 2 twisting automorphisms of GF(q). In Liebler's planes, B
a+ 1
1 and x
= 6
1 has no solutions in GF(q).
PROBLEM 9.1: (a) Determine the translation planes where kern
rr ~
rr
3
of even order q admitting SL(2,q)
GF(q).
(b) Same problem for odd order. (c) Same problem for even or odd order but with no restriction on the kern. Problem
(e)
at least would require representation theory.
The planes of Fou1ser-Ostrom have order q4 and admit SL(2,q).
Some have
2
kern GF(q ) and one has kern GF(q). PROBLEM 9.2: (a) Determine the translation planes of order q4 admitting SL(2,q) with kern
~
2
GF(q }. (b) Same problem with kern
~
GF(q).
(e) Same problem with no restriction on the kernel.
Translation planes that admit SL(2.pr)
507
Obviously, the picture is clear on how one might proceed, clicking the exponent i in qi up to one at a time while the plane of order qi admits SL(2,q). Probably, (9.2) is too hard.
But, (9.1)(a) is probably within reach, at least with
some slight modification such as adding (as Liebler does) the loo-orbit structure and a particular group fixing an invariant subplane pointwise. PROBLEM 9.3: Let
TI
be a translation plane of order pr that admits S ~ SL(2,ps) and
S acts irreducibly.
(a) Is there an invariant Hering or Ott-Schaeffer subplane (if r/2 kern ~ GF(p ), see (4.1) and (4.4))? (b) If there is an invariant Hering or Ott-Schaeffer subplane, must r/2 kern ~ GF(p )? (c) If there is an invariant Hering or Ott-Schaeffer subplane, is the plane TI
Hering or Ott-Schaeffer?
ADDED NOTES - ADDED PLANES 2.17 Kantor has recently constructed a class of translation planes of order p6r admitting GL(2,p2r) where the p-elements are elations. least pr(pr+l) nonisomorphic planes of each order. 6r
There are at
2.18 There is also a plane of order 43 admitting SL(2,4) which comes from the work of Dye and Kantor. BIBLIOGRAPHY 1. 2. 3. 4. 5. 6. 7. 8.
J.L. Alperin, Projective modules for SL(2,2 n) (to appear). R. Brauer and C. Nesbitt, On the modular characters of groups, Ann. Math., 42 (1941), 556-590. V. Dempwolff, Einige Translationsebenen der Ordnung 16 und Ihre Kollineationen (to appear). V. Dempwolff, Grosse Baer-Untergruppen auf Translationsebenen Gerader Ordnung (to appear). R.H. Dye, Partitions and their stabilizers for line complexes and quadrics, AnnaZi di Mat., (4) 114 (1977), 173-194. P. Fong and G. Seitz, Groups with a (B,N)-pair of rank 2, Invent. Math., 21 (1973), 1-57. D.A. Foulser, Baer p-elements in translation planes, J. AZgebra, 3 (1974), 354-366. D.A. Foulser, Derived translation planes admitting affine elations, Math. Z., 131 (1973), 183-188.
508
9.
10. ll. 12. 13.
14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.
N.L. Johnson
D.A. Foulser, A translation plane admitting Baer collineations of order p, Ai'clz. ,'·;.;;tri., 24 (1973), 323-326. 2 D.A. Foulser and N.L. Johnson, The translation planes of order q that admit SL(2,q) as a collineation group T, J. Algebm (submitted). D.A. Foulser, The translation planes of order q2 that admit SL(2,q) as a collineation group. II, J. Geometry (submitted). D.A. Foulser, N.L. Johnson and T.G. Ostrom, Characterization of the Desarguesian and Hall planes of order q2 by SL(2,q), Inter. J. Math. Sci., (submitted) . Ch. Hering, A new class of quasifields, Math. Z., 118 (1970), 56-57. Ch. Hering, Uber Translationsebenen, auf denen die Gruppe SL(2,q) operiert. Atti del Convegno di Geometria Combinatoria e sue Applicazioni. Perugia: 1st. t1at. Univ. Perugia, 259-261. Ch. Hering, On shears of translation planes, Abh. Math. Sem. Hamburg, 37 (1972), 258-268. Ch. Hering, On projective planes of type VI. Atti dei Convegni Lincei 17. Teorie Combinatorie (1976), 30-53. Ch. Hering and C.Y. Ho, On free involutions in linear groups and collineation groups of translation planes. Trabulho de r·latematica No. 129, Funda~ao Universidade de Brasilia (1977). N.L. Johnson, The geometry of GL(2,q) in translation planes of even order (and correction), Internat. J. Math. & Math. Sci., 1 (1978), 447-458. N.L. Johnson, The translation planes of Ott-Schaeffer, Arch. Math., 36 ( 1980), 183-192. N.L. Johnson, The translation planes of Bruck type {l }, Arch. Math., 26 (1975), 554-560. N.L. Johnson, Addendum to "The geometry of SL(2,q) in translation planes of even order", Geom. Ded. (submitted). N.L. Johnson, On the construction of the Dempwolff plane of order 16, Geom. ,'"Jed. (submi tted). N.L. Johnson, The translation planes of order 16 that admit SL(2,4), J. Cornb1:natorial Theory (submi tted). N.L. Johnson, The translation planes of order 16 that admit nonso1vable co 11 i nea ti on groups, Geom. Ded. (submitted). N.L. Johnson and T.G. Ostrom, Translation planes of characteristic two in which all involutions are Baer. J. Algebra (2) 54 (1978), 291-315. N.L. Johnson and T.G. Ostrom. Translation planes of dimension two and characteristic two, G6om. Ded. (to appear). N.L. Johnson and T.G. Ostrom. The geometry of SL(2,q) in translation planes of even order, Geom. Ded. 8 (1979), 39-60. N.L. Johnson and T.G. Ostrom, Translation planes with several homology or elation groups of order 3, Geom. Ded., 2 (1973). 65-81. W.M. Kantor, Spreads, translation planes and Kerdock sets. I. (to appear). R.A. Liebler, Combinatorial representation theory and translation planes (to appear). H. LUneburg, Translation Planes, Springer-Verlag: Berlin-Heidelberg-New York, 1980. H. LUneburg, Charakterisierungen der endlichen Desarguesschen projektiven Ebenen, Math. z., 85 (1964), 419-450. T.G. Ostrom, linear transformations and collineations of translation planes, J. Algebra, 14 (1970), 405-416.
Translation planes that admit SL(2,pr)
34. 35. 36. 37. 38. 39. 40. 41.
509
T.G. Ostrom, Elations in finite translation planes of characteristic 3, Abh. Math. Sem. Hamburg, 41 (1974), 179-184. T.G. Ostrom, Translation planes of dimension two with odd characteristic, Ganad. J. Math., (5) 32 (1980), 1114-1125. T.G. Ostrom, A class of translation planes admitting elations which are not translations, Arch. Math., 21 (1970),214-217. V. Ott, Eine neue Klasse von Trans1ationsebenen, Math. z.,143 (1975), 181185. B. Segre, On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two, Acta Arith., 5 (1959), 315-322. H.J. Schaeffer, Trans1ationsebenen, auf denen die Gruppe SL(2,pn) operiert. Dip10marbeit TUbingen, 1975. 1.0. Suprunenko and A.E. Za1esskii, Classification of finite irreducible linear groups of degree 4 on fields of characteristic p> 5. Inst. Mat. Akad. Nauk. BSSR, preprint No. 13 (1976). M. Walker, On translation planes and their collineation groups, Ph. D. Thesis, Univ. London, 1973.
The University of Iowa Iowa City, Iowa 52242
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Annals of Discrete Mathematics 18 (1983) 511-518 © North-Holland Publishing Company
511
NONDESARGUESIAN PLANES, PARTIAL GEOMETRIES, STRONGLY REGULAR GRAPHS AND CODES ARISING FROM HYPERBOLIC QUADRICS William M. Kantor
ABSTRACT Spreads of hyperbolic quadrics are used to construct translation planes, partial geometries, strongly regular graphs and codes, all having a rich geometric structure. 1. INTRODUCTION A hyperbolic quadric Q+ in PG(2n-l ,q) has (qn_ l )(qn-l+ 1 )/(q_l) points, and contains n-l-spaces each having (qn_l)/(q_l) points.
A $ppead of Q+ is a family
r of qn-l+ 1 subspaces of Q+ of dimension n-l partitioning the points of Q+.
A
spread can only exist if n is even (when n is odd, Q+ does not even contain 3 pairwise disjoint n-l-spaces).
The only known examples occur when n=2, when q is
even, or when n=4 and q;: 0 or 2(mod3). In this paper, we will summarize some of the ways spreads of hyperbolic quadrics have been used recently. 2. TRANSLATION PLANES AND THE CONSTRUCTION OF SPREADS If X is any subspace of PG(2n-l,q), then Xl will denote its polar with respect to Q+.
Consider a spread r of Q+, and assume that q is even.
x ~ Q+, and form the family
i
n r = {xl n wlw
n-2-spaces partitioning the quadric xl n Q+.
E
r}; clearly, this consists of
Since q is even, x
a PG(2n-3,q) equipped with a symplectic (or null)
Fix any point
polarity.
E xl
and xl Ix is
If r(x) denotes the
projection of xl n r into xlix, then r(x) is a spread in the more usual sense: qn-l+1 subspaces of dimension n-2 which partition the points of xl/x.
Consequent-
ly r and x determine a translation plane A(r(x)) arising from a symplectic spread. Conversely, suppose that nand q are even, and that spread of PG(2n-3,q). [4], Dye [5]).
~,
is a symplectic
Then~' arises from some Q+, r and x, as follows (Dillon
We can regard PG(2n-3,q) as our former xl/x, related to Q+ as be-
512
W.M. Kantor ;.)-1
fore. Xl
(I
Each of the j'
+1 members of
~,
is the projection of a unique n-2-space of
Q+, and the resulting family ~" of n-2-spaces partitions
xl
Q+.
II
Fix one of
the two families of ~-l-spaces of Q+; each n-2-space of Q+ is in a unique member of this family.
Thus, r" lifts to a family ~ of qn-l+l of these n-l-spaces, any
two having at most a point in common.
Since no two of these n-l-spaces can have
exactly one point in common (as n is even), l consists of pairwise disjoint subspaces.
Consequently, r is a spread of Q+.
The preceding construction produced an essentially unique spread of Q+ from 1
a symplectic spread of x Ix.
In particular, l(x) essentially determines l (where
"essentially" means that the symplectic geometry on / Ix can arise from several quadrics in PG(2n-l,'7), and that we singled out one of the two families of n-lspaces of and x
1
+
~).
However, more is true.
be pOints off Q+.
Let l and
~
+
1
be spreads of Q , and let x
Then any isomorphism from A{l{x»
collineation of PG(2n-l ,q) sending Q+ to itself, ~ to l In particular, each collineation of
A(~(x»
1
to A{l (x » induces a 1
and x to x
1
1
(Kantor [6]).
is the product of a perspectivity with
axis at infinity and a collineation of PG(2n-l,q) preserving Q+, land x.
Thus,
the collineation groups of many translation planes can be found simultaneously once the group c{r) of collineations preserving Q+ and
L
is known.
Several spreads l are described in Kantor [6,7], and some of the resulting n-l
translation planes of order q
are studied in detail.
q=2, only desarguesian planes occur.
When n=2, or n=4 and
In all other cases, new planes are obtained.
Some of the planes are uninteresting, but others have collineation groups behaving in unusual manners (such as flag-transitively, or behaving as decribed in Johnson's paper in these proceedings).
Complete arcs and dual ovals in all of
these planes were found by Thas [13] . The Simplest example of a hyperbolic spread is obtained as follows.
The
desarguesian plane AG(2,qn-l) arises from the unique spread of pc(l ,qn-l), which is trivially symplectic.
This produces a symplectic spread l' in PC{2n-3,q), and
hence (if nand q are even) a spread L of a hyperbolic quadric in PC(2n-l ,q}. This spread is called desarguesian, for obvious reasons.
It was found by Dillon
[4] and Dye [4]; G{l} was determined by Dye [5] and Cohen and Wilbrink [2].
As we
will see in §6, an "affine" version of this spread was discovered much earlier by Kerdock [ 10) . Other hyperbolic spreads arise from the hermitian curve in PC(2,q2), from triality, and from field changes generalizing that of the preceding paragraph.
Geometric structures arising from hyperbolic quadrics
513
For coordinate descriptions of many of the known examples we refer to Kantor[6,7). It seems unlikely that hyperbolic spreads can exist when n > 4 and q is odd. On the other hand, if q is even and fixed, the number of inequivalent hyperbolic spreads probably
+00
as n+oo.
3. MORE TRANSLATION PLANES; OVOIDS Let Q+ be a hyperbolic quadric in PG(7,q), where q is now even or odd. Consider a spread ~ of Q+. Q+.
Then ~ belongs to one of the families of 3-spaces of
Let T be any member of that family not in~.
empty or a line.
Set T n
~
= {T n
wlw E
~
and T n
If W E ~ then T n W is either ~
is a line}.
spread of PG(3,q), and hence determines a translation plane of order q2. translation planes obtained from
~
~
Then T n
is a
(The
3
as in §2 when q is even have order q , not q 2 .)
These translation planes are studied in Kantor (8). Let, be a triality map. Then, cyclically permutes the following three sets: the points of Q+, and the two families of 3-spaces on Q+.
At the same time,
, sends lines of Q+ to lines of Q+, while preserving incidence between lines and It follows that ~' is either another spread, or
both points and 3-spaces of Q+.
else consists of q3+ l pairwise noncol1inear points of Q+. Let Q+ be a hyperbolic quadric in PG(2n-1,q). Thas (14)) to be a set n of
q
_1
An ovoid of Q+ is defined (by +
+1 pairwise noncollinear points of Q.
count shows that each n-1-space on Q+ contains a unique point of n.
A simple
It follows
that, if x E Q+ - D, then xl n n projects onto an ovoid D(x) of the obvious . 1n . x1;x. qua dr1C
If n > 4, ovoids probably do not exist, but this has only been proven in PG(2n-l,2) (Kantor [9, (4.3))).
If n=3, ovoids correspond (under the Klein corre-
spondence) to spreads in PG(3,q), and hence to translation planes of order q 2 . n=4, ~ is a spread, and ~' is not a spread, then ~' is an ovoid n.
If
Moreover, if T is as before then T' is a point x, and T n ~ and n(x) are related by the Klein correspondence. with D(x) than T n
~,
When dealing with coordinates, it is easier to work
since points of n require 8 coordinates while 3-spaces in
The translation planes A(n(x)) are studied for
are more complicated to describe. all known
~
recover n or
in Kantor (8). ~
~
Unlike the situation in §2, it is not clear how to
from A(D(x)); moreover, some co1lineations of the plane need not be
related to automorphisms of n or
~.
W.M. Kantor
514
4. STRONGLY REGULAR GRAPH Let ~ be a spread of a hyperbolic quadric Q+ in PG(2n-l,q), where n ~ 4. Let 11 be the set of all hyperplanes of members of ~, so that 1111=1~I(ql1-1)/(q-l) is the number of points of Qt.
If
X, Y E
11 and
X
F
Y,
write x-y=-iny F O.
Then
(11,-) is a strongly regular graph having the same parameters as the col linearity +
+
graph (Q ,1) of Q (Kantor [9]).
If n=4, these graphs are isomorphic (an
isomorphism being induced by triality). isomorphic.
However, if
11
> 4 they are probably never
This is known if q=2 (see [91); the proof uses the nonexistence of
ovoids of Q+ when q=2.
The graphs are also not isomorphic when ~ is the
desarguesian spread defined in §2; this was proved in [9) by brute-force calculations. If
WE
~ 1et ;/' be its set of hyperp 1anes.
partition of n into cliques. ( 3 is isomorphic to a trace-space given by a projective space and a quadratic set. In
§
3 we consider self-dual ovals O.
group f(O) in a dual way.
Here we can define a second oval
We shall give examples of projective planes (P,B) with
self-dual ovals 0, where the groups r(O) and r(O) are different.
By Theorem 1 any
example where one of the groups r(O) or r(O) is sharply 3-transitive and (P,B) is not pappian can be derived from a projective plane over a commutative field F,
The projectivity groups of ovals and of quadratic sets
521
where 0 is a conic, by replacing the lines which have an empty intersection with 0 by other curves. tions (see
§
To obtain such examples the field F has to fulfil suitable condi-
3).
Theorem 2 of
§
4 gives an answer to the second question and Theorem 3 of
§
5
deals with the group r(Q) of a quadratic set Q consisting of two distinct lines of a projective plane.
1. DEFINITIONS Let (P,B) be a projective space with the point set P and the line set Band Q + ~ a subset of P.
For the structure (P,B,Q) we define the following types of
lines: B:={LEB
LeQ},
B;== {L E B
IL n Q I =
i}
for i E
]'j
The set Q is called a quadratic set, if the following axiom holds: 1 B = Bo U Bl U B2 U Boo In the same way we can define a dual quadratic set
Q
Q.
restrict ourselves to the case of a projective plane.
In this paper we will Then
0 is
a subset of B
characterized by the axiom Pl U P2 U Poo If [p] := {X E B : p E X} then Poo:= {p E P: [p] eO}and Pi := {p E P:I[p] nQI=
Q l'
P = Po
U
n.
A subset Q e P is called a self-dual quadpatic set if for Q and O:=B l the axioms Q 1 and Q l' are valid respectively. Special types of quadratic sets Q are the ovals in projective planes which are characterized by the condition: Q 2 For any q E Q we havel [q] n Bl I = 1.
We shall write [P]l := [p] nB l · For an ovalO we can identify the point-set 0 with the line-set
0 .-
Bl by the
bijection
o --~
0 = Bl ; x ---+[x] n Bl , and we have Boo = By these definitions we obtain:
~
.
If 0 is a self-dual oval of a projective plane (P,B) then O:=B l is an oval in the dual projective plane (B,F) with P:={[p] : pEP}. ~
In the case of a projective plane over a commutative field with char (F)F2,
522
H. Karzel and M. Marchi
any ellipse 0 is a self-dual oval. For any projective space with a quadratic set (P,B,Q) we define now the projectivity group of Q. For any pEP \ 0 let
p be
the permutation of 0 defined
by (
I 0 --... ;
p
Q (
i x ___~x \
lx'
p,x n Q = x
if
= p,x
n
0 \ {x}
if
p,x n Q + x.
\
By p,x we denote the line L E B with p,x E L.
The permutation group
,(Q) := generated by all these mappings is called the pY'ojectivi~i!
group of the quadi'atic set Q.
If
0 is
a dual quadratic set of a projective plane then any line L E B \ Q
defines a permutation
I Q- ---... 0(
L
J
(
Ix
I
I x ---~ I
IX'
if :=
[ X n L) n 0 = X
[X n LJ n Q \ {X}
if (X n LJ n Q + X,
\
and all these maps generate the projectivity group r(o) := .
Any
self-dual quadratic set 0 has the two projectivity group r(O) and r(o) with Q = B . In the case of a self-dual ovalO (cf. (1.1)) we will consider r(Q) as a l permutation group of O. For n
IN a permutation group (M,r) is called n-transitive if for any two n n-tuples (Pl, ... ,Pn),(ql, ... ,qn) E M with I{Pl, ... ,Pn}1 = l{ql, ... ,qn}1 = n there E
exists ayE r with Y(Pi} = qi for i E fl,2, ... ,n}.
We say that a permutation
group (M,r) has the pY'opeY'ty (P ) if a map y E r fixing n distinct points of M is n the identity. Moreover we say that a permutation group (M,r) is sharply n-transitive if it is n-transitive and has the property (P ). n
2. OVALS WITH A SHARPLY 3-TRANSITIVE PROJECTIVITY GROUP In this section let 0 be an oval of a projective plane (P,B) with IO!
~
6 and r:= r(O}. 2.1
First we show the following.
Let p,q E P \ 0 be any two distinct points; then
P+ q.
The projectivity groups of ovals and of quadratic sets
523
PROOF: Let x be any point on 0; then x' := pix) E p,x, x" := q(x) E q,x. x
f
p,q we have x'
If
f x".
From this proposition we have: 2.2 There exists at most one point pEP' 0 such that p = id.
We have to distinguish the two cases: I) for all pEP \ 0 we have p
f
i d,
II) there exists exactly one point kE P' 0 with k = id; the point is then called nueZeus of O.
Henceforth by a,b we shall mean any two distinct points of 0; we denote: ~ a, b:= {p : p E a,b, p
f
a,b} and
f(a,b):= {y E r(O) : y(a) = a, y(b) = b} .
.£.:l
For any poi nt pEP' 0 we have:
p E a,b' {a,b} ~ pia) = b
P E ~ a, b'
2.4 The permutation set (0' {a,b},
~
a, b'
~
a, b) is transitive.
PROOF: Let x,x' EO' {a,b} be any two distinct points; if yEO' {a,b,x,x'} then we denote {r}:= x,y n a,b, {s} := x' ,y n a,b.
Since r,s
~
{a,b} we have
sr(x)=x'.
By this proposition and by ~ a, b'
~
a, b ~ r( a, b) we obtain:
2.5 The permutation group (0 '{a,b}, r(a,b)) is transitive. 2.6 The projectivity group (O,r) is 3-transitive. PROOF: Let x,y,z,x',y',z' E 0 with l{x,y,z}1 = l{x',y',z'}1 = 3. assume x = x'.
Since 101
~
By (2.5) we can
6 there is an element u E 0 '{x,y,y'} and (by (2.5))
524
H. Karzel and M. Marchi
an a E '( x,u ) such that a(y) = y'. Therefore we can now suppose x = x' and y = y'. Again by (2.5) there exists a 8 E f( x,y ) such that 8(z) = z'. Let us assume from now on that (O,r) has the property (P ); then we have: 3 -2.7
il a, b' ~ a, b' ~ a, b
PROOF: Since (~a,b)3 q,r,s E a,b \ {a,bL
~a,b
.
~ ~a,b'
We have to prove (6a~b~3_~ ~a,b' Let For x E 0 \ {a,b} let x' := s r q(x) and {u}:= x,x' n a,b .......
(where x,x'
""'-'
................
if x = x'); then the projectivity u s r q fixes the points 1 ............... -- ...... a,b,x and hence, by (P ) u s r q = id. that means u = s r q. 3 :=
[xl
2.8 The permutation group (0 \ {a,b}, f(a,b)) is regular and commutative. PROOF: Since the subset ~ a, b' 6a, b of f( a, b) operates transitively on 0 \ {a,b} (by (2.4.)), f(a,b) is regu1~r_b: ~P3); hence f(a,b) = 6a ,b . ~a,b' To prove the commutativity of r( b) let p,q,r,s be involutions in 6 b; then by (2.7) we have - - -
a,
_ _ _ _
_ _ _ _
_ _
~, -1-
--1 --1 --1 -
(p ____ q r) E 6a, b and hence: (p___ q)( r s) = (p q r) s = (p_ q___ r) s -----1 = (r q p )s = r q p s. Again by (2.7) q p s E 6a, b and hence: r q p s = r(q p s) = (r
s)(p q). ~
Let a E r be any involutorial transformation (a
+ id.).
Then there
exists exactly one point pEP \0 such that p = a. PROOF: By (P 3 ) there are x,y E 0 with x + a(x),y, a(x) + y and a(y) + y. For {p} := x,a(x) n y,a(y) the map p a has the three distinct fixed points X,a(X),y
-
and hence by (P ) p a 3
=
id. i.e. -p
=
a.
By (2.1)
2.10 For a EO and p E P\ 0 we have: p(a)
P is unique. =
a" p E [all'
For a E 0 let 6 := {p : pEP \ 0 with p,a n 0 = {a}}. a Viith the same arguments as in the proof of (2.4) it follows: 2.11 - The permutation set (0 \ {a}, 6a • 6a ) is transitive •
The projectivity groups of ovals and of quadratic sets
Let us denote by J the set of all involutions in r(O).
2.12 {p {p
525
By (2.9) we have:
pEP \ O} = J if 0 has no nucleus pEP \ O} = J U {id.} if 0 has a nucleus k.
Let now 00,0,1 be three distinct points on 0 and
:= (00],
k
n
[0]1'
Since (O,r) is
sharply 3-transitive there exists a KT-field (F,+,·,a) (see [4], [8]) such that i)
O=FU{ooL
ii)
0,1 are the neutral elements of (F,+) and (F'~,.) resp. (where F'~:= F \ {O}).
iii)
For a,b E F, b
+° let
(
[a,b] : F U {oo}
---->
F U {oo}; x
x 4 oo ---->t00 + bx for for x = oo 1
Then roo = {[ a, b] : a, b E F1\ b iv)
Let.: F U {oo}
--->
+ o}.
l
F U {oo} with .(x) .- a(x) for x E F''', .(0) = 00 and
.(00) = O.
Then r = roo U JiF [c, 1] .r oo ' Let F+ := {c+ : c E F} with c+ := [c, 1] and Joo = roo () J.
v)
(k vi)
Then F+=Joo ·kU{id.} is the involution fixing 0 and oo, or the identity if 0 has a nucleus)
(F*,·) and r(O,oo) are isomorphic. By (2.8) r(O,oo) is commutative; hence (F,+,·) is distributive by (vi), and
therefore, by [9], a commutative field. -1
By the theorem of Benz-Elliger ([2]) we have moreover a(x) = x
for all
)t,
X E F and therefore r = PGL(2,F). . ax + b For a E PGL(2,F) wlth a(x) = CX + d we have ad - bc
°
aEJ ~ a + d = for char(F) a E J U {id.} ~ a + d = 0 for char(F)
+
°and:
+ 2, 2.
Hence together with (2.12) we may identify the point set P \ 0 with the set of all points F*(a,b,c) E F3*/F* of the projective coordinate plane R(F) over the commutative field F for which we have /+bc 3'~
,':
3'~
+ 0.
This bijection between P \ 0 2
and F IF":\C with C := {F (x ,x 2 ,x ) E F IF'~ : Xl + x2x3 = O} is extended to a l 3 ,': 3* ,~ 2 bijection w : P --> F IF* by w(oo) = F (0,-1,0) and w(t) = F (t,-t ,1) for t E F. Now we prove:
526
H. Karzel and M. Marchi
2.13
For L E Bl
PROOF: Let p 1. Case:
LE
pEL" b =
B 2
B2 the set w(L) is a line of
U
F (x, ,x ,x ) E P \ O. 2 3 • Let {a,b} := L n 0 with a
p( a)
x,a
+
=
x2
+
00.
IT
2
(F).
By (2.3) we have:
(, (*)x,(a+b)+x -x a b = 0, for b + 2 3
1
00
.. )
for b = x3a - xl (**)x l -x 3a = 0, Since oo(a) = F*(a,-a 2,1) and w(b) = F*(b,-b 2 ,1) or w(a) and w(oo)
00
F*(O,l,O) are
3'~
the only other points of F IF''; satisfying the linear equation ("') or (,';"') respectively, will is a line of IT 2(F). :;. C,we;: L E B . Let {a} := L n O. l pEL"
a = pta) .. J )
By (2.11) we have:
2 (*) 2x l a+x 2 - x3 a
l ('' ' ') x3 ° =
Since w(a) 3 F "'1 F'"
= FA(a,_a 2 ,1) or w(a) = w(oo)
=
°for a +
00
for a F(O,l,O) is the only remaining pOint of
satisfying U') or (""") resp., will is aline of
IT
2(F).
With these lemmas we have proved: 2.14 Let 0 be an oval of a projective plane (P,B) with a sharply 3-transitive projectivity group rIO). P such that (P,B'o By (2.14)
U
Bl
U
Then there exists a subset B'o of the power-set of
B ) is a pappian projective plane in which 0 is a conic. 2
Theorem 1 is proved for the projective case.
For the attlne case
let 'A(O) be the oval group generated by perspectivities with centers in the affine plane, and frO) the oval group generated by perspectivities with centers in the extended projective plane.
Then one shows that fA(O) is 3-transitive and that
the condition (P 3) for fA(O) implies (P 3 ) also for f(O). completely.
So Theorem 1 is proved
3. SELF-DUAL OVALS WITH DIFFERENT PROJECTIVITY GROUPS In this section (P,B,O) will always be a projective plane with a self-dual ovalO.
We will give examples of self-dual ovals 0, where one of the groups frO)
or rIO) fulfills condition (P 3 ), the other does not. By the results of § 2 we have:
The projectivity groups of ovals and of quadratic sets
527
Let r(O) fulfill condition (P 3 ) and let F be the corresponding commutative field. Then there exists a line set B' such that: 3.1
a) (P,B') is the pappian projective plane of F; and 0 an ellipse, b) For any two distinct pOints p,q E P2 u 0 we have p,q n P 2 p,qEp,q'EB'.
=
p,q' n P , where 2
+ 2 and
Now let F be a commutative field with char (F)
0 the ellipse of the 2 projective plane over F given by the quadratic form q(x) = x + x x . Let 1 23 f(x,y) = q(x+y) - q(x} - q(y) = 2x l Yl + x2Y + x3Y2 be the corresponding bilinear 3(2) (2) 2 ,', form, and F/F(2) := {x.F : x E F} with F := {x : x E F } the set of
quadratic classes.
Then the point set P is decomposed into the subsets := {F*' x: q(x) E a.F(2}}, and we obtain:
Pta} := p(a'F(2» 3.2
0 = P = P(O), P = P(l), Po =
3.3
For any L E Bl we have L C P2
l
2
{P(x) : x E F'': with x
U
U 0
3.4 For any L E B2 we have L n P(r)
= P(l)
+ 1/1
for all rEF.
*
3.5 For all L E B we have L n P2
1(F(2»}.
U P(O}.
PROOF: For {('a, F b} := L n 0, c := f(a,b) (+ 0) we have q(a + c c- l rf(a,b) = r; hence F*(a + c-lr • b) E L n P(r). 6
$
-1
r b)
+1/1.
PROOF: By (3.3) and (3.4) we have LEBo to consider.
Since for A E Bl , An L ~
1/1
we have An L c P by (3.3). 2 3
1,
3.6 Let L E Band u E F with L = {F x E P L C p -=-q(U) 2
=-1
(F(2»
f(u,x}
=
O}.
Then:
F is pythagorean. ,,:
(2),':
PROOF: By (3.5) there is a point F a E L n P2 ; hence q(a) E F . Let F bEL with f(a,b) = O. Then, by f(a,u) = f(b,u) = f(a,b) = 0 and q(a)·q(b)·q(u) + 0, we
=
have q(a).q(b).q(u)E(-1)'F(2); hence q(u) (-l)·q(b). For any point Fi'.x E L we 2 2 have x = xla + x b and q(x) = Xl q(a) + x2 q(b). Hence: L C P2 ~ V Xl ,x 2 E F with 2
528
H. Karzel and M. Marchi
(Xl ,x Z) ~ (0,0) we have both pythagorean
xl q(a) + xl q(b)
E
F(Z) and F is
q(u) '" -1.". F is pythagorean.
The following conditions are equivalent:
~
o ' ~ ALE B0 => L C PZ ' b) F is an euclidean field.
a) B i
PROOF: let l LEBo l C P
=>
z
=
{F'" x
E
P: f(u,x)
a}.
=
Then by (3.Z):
q(u) Ef F(Z) u {a} and by (3.6): q(ll) E - F(Z)", F(Z) + F(Z) C F(Z) (2)
Hence a) is equivalent to F
.'. + (', (..'. \ F(2)
i. e.
F is euclidean. For F(Z) ~
F'" we have Po =
=
if>
and Bo
= if>
and hence:
Let (P,B,O) be a projective plane with a self-dual oval 0 such that one
of the groups riO) and riO) fulfills (P ). 3 field F, then (P,B) is pappi an.
If F(Z) = F* for the corresponding
To obtain examples (P,B,O) such that (P3) is valid for riO) but not for riO) we have to start from a projective plane over a field F with char(F)
1 F"',
F(2)
~
Z and
and by (3.1) we have to replace the intersections L n Po between lines
L and the interior Po of the ellipse 0 by other subsets of Po'
If F is an
euclidean field this can be done with an easy method: With the SUbstitution
= YZ - Y3 we obtain q(x) " Yl 2+y~
Xl " Yl' Xz = Y2
+ Y '
fine plane {(x,y)
:=
F"'(x,y,l)
l
+
(1 = ,;
3
((x,y) E F2 :
x3
i
x,y
= l}.
E
- y;, and in the af-
F} the ellipse 0 has the form
The line U of infinity (spanned by ("(1,0,0) and
F (0,1,0)) belongs to Bo; hence by (3.7) U C P and 0 C P \ U. 2 Now we consider also a Moulton plane over F where the x-axis is the bending line.
For two distinct points p,q
B2
P let p,qm be the joining line of p,q in the
BZ let LZ := (P u 0) n L, {ll' lZ} L no, 2 :" Po n (1 1,1 2 ), and L' :" L~ u LZ' Then (P, Bo U B1 u B ,0) with 2 :=' [L' : L E B } is a non-desarguesian projective plane with a self-dual oval Z
Moulton plane. L~
E
For any line l
E
--m
riO). 3 Another example (P,B,O) where r(O) fulfills (P3) but not riO) is the
0, where riO) fulfills (P ) but not
projective plane given by Hilbert (1899) in the first edition of his "Grundlagen
529
The projectivity groups of ovals and of quadratic sets
der Geometrie".
This was the first example of a non-desarguesian projective plane
Hilbert chose an ellipse in the euclidean plane over the reals and replaced the segments of secants by sections of suitable circles.
Also the example of T.
Vahlen (1905) in his book "Abstrakte Geometrie" of a non-desarguesian projective plane has this property.
4. THE PROJECTIVITY GROUP OF A SELF-DUAL OVAL In this section let 0 be a self-dual oval a projective plane (P,B).
Then we
may consider the two projectivity groups (O,r(O)) and (O,r(O)), which are 3-transitive by (2.6) and (1.1).
Now let us assume that (O,r(O)) fulfills the condition
(P 3) hence that (O,r(O)) is a sharply 3-transitive permutation group.
By theorem
1 there exists a subset B; of the power set of P such that (P,B; U Bl U B ) is a
2
pappian projective plane where 0 is a conic.
Since this conic 0 is self-dual, the
tangents cannot pass through a common point.
Therefore the corresponding commu-
tative field F has a characteristic different from 2.
By
§
2 every point p of P
is represented by F,·,(Pl,_P2) with (Pl,P2,P3 + (0,0,0). The determinant P3' Pl 2 _(X + x x ) is a quadratic form and the corresponding bilinear form 2x Y + l l 23 1 + x Y3 + x Y defines a bijective polarity n: P---~ B; U Bl U B2 with n(P ) = 6 3 2 1 2 l and n(P2) =B2 . For any pEP \ 0 the involution p is induced by the projective reflection of the projective plane (P,B;
Bl
U
U
B2) with center p and axis nIp).
Therefore we have: := of rIO) is also generated by the set {L : L E B2 } i.e. ~ is a subgroup of rIO). ~
The subgroup
~
By calculation one obtains:
{F
f,
(Xl ,X 2 ) x3 ' - x1
E
PGL(2,F): x x - x 1 4
X
2 3
(2)
EF
U
(2)
(-1) F
}
•
530
H. Karzel and M. Marchi
As we saw in
2 (cf. (2.12)v)) the addition group (F.+) of the correspond-
§
ing commutative field (F,+,') is isomorphic to the group F+ = J oo . k. + tion we obtain F = {p q ; p,q E [001 1 /\ p,q + oo}. ~
Since ([ 0011\ roc))
By calcula-
~
Pz we have
C
+ ++ + 4.3 F uF 'o=F UF (-1)'
(a
+
+ aEF)u{a(-l)'
aEF)
F+ u F+ (-1)' ~ ~.
Since the polarity TI interchanges the points of the tangent with the lines containing
00
we have TI([ooll\oo) = Bl
n [001
[00 1
1 and by (4.3);
through
00
The multiplication group (F*,,) can be considered as a subgroup of
'I
);= of C(M ) n which satisfy H -2(X,E] 0, where E := (0, ... ,0,1) is called the antipode.
*
=
Equations of the form (X,A) = 0 for A
*0
in C(M ) represent quadrics, null cones n
or hyperplanes in M (see (5] for details); if (A,A] = 0 and (A,E) t 0, for examn
p1e, we obta in the nu 11 cone with vertex E Mn. ,~'onformal Minkowski space Mn
is obtained by allowing paints with H = 0,
i.e. by adjoining to M those <X>c C(M ) satisfying [X,X) = [X,E) = n
n
o.
Since
(E,E) = 0, we say that the added points form the antipodal null cone, or the cone (~t
infini ty .
We set e : = <E >E M and C(M ) n
n
= C(Mn ).
Note that X E C(M ) repn
presents the finite point x E M iff X cr(x,(x,x)). n
We shall be interested in the extensions of null lines in M to n
Mn ,
particu-
lar1y in those properties summarized in the following lemma.
LEMMA 3.1: a) Le;; a +be a null line in Mn' and define A.- (a,l,(a,a)), p"':= (k,0,2(a,k)) E C(M).
7'hen
n
U ,
Cl
'ih:) a += {<X> IXE}
COniJel'seZy, for any totally isotropic 2-space W in C(f\) with
E Ef W,{ <X> I X E W, [X,E) t)
* O}
For n:;,[Z lines a
1
is a null line in M . n + , a + in M , define 1
2
p~' .- (k ,0,2(a ,k )), p~ := (k ,0,2(a ,k )). 1
1
1
2
2
2
2
n
Then the lines are paralleZ iff
<Jl'7,P;,E> is a totai.lu isotropic 2-space in C(~)
PROOF: a) Parts i) and ii) follow from direct calculation.
For part iii),
oo
X E \
iff X crAHP'" for some \ E R, i.e. iff Xcr (a+Ak, 1 ,(a,a)+2A(a,k)). This last holds iff x = a + \kEa +for x =<X>. Conversely, choose distinct
A,B in W with [A,E) = [B,E)
a = E Mn' A - B = (k,O,w) for some k E Mn' wEIR. a + ={ <X> IX E W, [X,E)
t
0, and set
Then k is null and
* OJ.
b) From a)i) is a totally isotropic 2-space in C(M ). n are para 11 e 1, then k = uk for some u E lR; thus poo _ a 1
1
P; = (0,O,2(a
l
If the 1ines
2
-a ,k 2
1
)) =
B E for some BE IR. i.e. p7 E.
ly, if P: = a P; + B E for some u,s E IR, then kl = uk
2
;
Converse-
hence the lines are par-
Conformal transformations of Min kows ki spacetime
alle1.
571
0 oo
We may think of the point
in the above lemma as the point at infinity of the line a +.
A totally isotropic 2-space W in C(Mn ) which does contain E satisfies [W,E] = 0; in this case, the set {<X>!X E W} C Mn may be thought of as a null line at infinity containing e, i.e. as a generator of the null cone at infinPart b) of our Lemma then states that parallel null lines in Mn intersect the same generator of the null cone at infinity. ity.
The isometries of its coordinate space induce the conformal transformations Since isometries of C(M n) preserve totally isotropic 2-spaces in C(M n), conformal transformations map null lines in Mn onto null lines. Furthermore, the
of Mn
conformal transformations which fix e (and thus map Mn onto itself) induce the similarities of Mn (see [5] for detail). The two lemmas which follow will be used in the proof of our theorem. LEMMA 3.2; For p,q
E
Mn with (p-q, p-q)
tion g1 of ~ such that g1(P)
* 0,
there exists a conformal transforma-
= e, g1(q) = o.
PROOF: Set P := (p,l,(p,p)), Q:= (q,l,(q,q)), Z:= (0,1,0) and,,:= -2[P,Q] = (p-q, p-q)
* O.
The linear mapping ..... <E,Z>given by p ..... E, Q ..... feZ is an
isometry, and is thus the restriction of an isometry G1 of C(M). n
The conformal
transformation 9 of M induced by G has the required properties.
0
n
1
LEMMA 3.3: For a,b
E
1
Mn with (a-b, a-b) > 0 and any t
E
Mn with (t,t)
exists a conformal transformation 9 of Mn such that 9 (a) 2
2
= 0,
9 (b) 2
1, there e and
PROOF: Set A:= (a,l,(a,a)), B;= (b,l,(b,b)), Z:= (0,1,0), T:= (t,l,l) and " := -2[A,B] = (a-b, a-b) > nates of (1/2)(a+b). 1
I
o.
Then C := A + B + "E gives the conformal coordi-
The linear mapping ..... given by 1
A ..... ,,2Z, B ..... ,,2E, C ..... 2,,2T, is an isometry, and is thus the restriction of an isometry G2 of C(M). n
The corresponding conformal transformation g2 of Mn
satisfies g2 (a) = 0, g2 (b)
=
e, and g2[Ha+b)] = t. .
Now, N(a,b) is that region bounded by C(a) and C(b) which contains !(a+b). It
follows that g [N(a,b)] is that region bounded by CeO) and the null cone at 2
572
J.A. Lester
infinity which contains t, thus g [N(a,b)] n M has the stated form. n
2
0
Fix t E Mn with (t,t) = 1, and denote by S that region of Mn bounded by C(O) and the cone at infinity which contains t. The set S := S n M has the important o
n
property that all null line sections of So have their points at infinity in S.
4. PROOF OF THE THEOREM (Suppose f(a) = fIb) for some a
First of all, f must be injective.
then a-b is null, and CIa) n C(b) is the null line joining a and b.
* b;
An open set
DC V containing a contains point d on CIa) but not on C(b), thus f(d) is on C{f(a)] but not on C[f(b)), which is impossible.) LEMMA 4.1:
POP
a,b E Mn with (a-b, a-b) > 0, suppose that an injective function
9 : N(a,b) .... Mn satisfies, fop aU x,y E N(a,b), (x-y, x-y) i'::en
= 0 iff (g(x) - g(y), g(x) - g(x) - g(y)) = O.
*
9 is the pestl'iction of a confopmal tpansformation of M • n
PROOF: From Lemmas 3.2 and 3.3, there exist conformal transformations 9 ,g
M such n
1 2
that 9 1(g(a)] = 0, 9 1(g(b))
for S as in
§
= e, g 2 (a) = 0, 9 2 (b) = e, and 92[N(a,b)]
S
4. -
-1
Define h : S .... Mn by h(x) = gl[g(g2 (x)] for all XES; then h(O) hIe) = e, so from *, XES is finite iff h(x) is. sections of S into null lines, thus for So := S
= 0 and
Furthermore, h maps null line
Mn, h takes null line sections of S into null lines in M , while preserving their points at infinity. Since h o n also maps sections of S with generators of the cone at infinity into like n
generators, Lemma 3.1 yields that the function ho := hiS maps parallel null line o
sections S into parallel null lines. o Any three parallel null line sections of So lying in an Artinian 2-p1ane are all intersected by another null line sections of So'
Their parallel images are
then also intersected by a null line; they must lie in an Artinian 2-p1ane. Therefore h maps Artinian 2-p1ane sections of S into Artinian 2-p1anes, and o 0 moreover, maps parallel Artinian 2-plane sections into parallel Artinian 2-p1anes, since such planes are determined by parallel families of null lines. Two Artinian 2-p1anes which intersect at a point pESo contain between them at least three null line sections of So through p.
The corresponding image 2-
Conformal transformations of Minkowski spacetime
573
planes cannot coincide, since then two of the null line sections would be mapped into a single null line, thereby mapping some pair of points x,y with (x-y,x-y)+ 0 into points satisfying (ho(x) - ho(y), ho(x) - ho(Y))
= 0,
and thus contradicting
Since any line section of So lies in two distinct Artinian 2-planes, and
1:
indeed, parallel line sections of So lie in two pairs of parallel Artinian 2planes, ho maps line sections of So into lines and preserves their parallelism. It now followd easily that ho is the restriction of a bijective linear transformation of Mn, which, since it maps C(O) onto itself, must be a scalar times a Lorentz transformation. This multiple of a Lorentz transformation must then be the restriction of a conformal transformation s of Mn which agrees with h on all of S, since both preserve the points at infinity of null line sections of So'
It follows that 9 is the restriction of the conformal transformation
-1
gl
0
S
--
0
g2 to N(a,b).
0
Since V is open, for any x E V there exist a,b,c,d in such that x
E
N(a,b)
C
N(a,b)
C
N(c,d) c V.
a conformal transformation.
From Lemma 4.1, fIN(a,b) is the restriction of
On non-empty intersections of two such N(a,b)'s, the
corresponding conformal transformation agree; thus, since they are analytic functions, they agree on the union of the two N(a,b)'s.
Since V is connected, all
conformal transformations so obtained agree, thus there exists a single conformal transformation c of Mn such that f
=c ' iV
ACKNOWLEDGEMENTS: This work was completed with the financial assistance of the Alexander von Humboldt Foundation.
BIBLIOGRAPHY 1. 2.
3. 4.
H.J. Borchers and G.C. Hegerfeldt, The Structure of Spacetime Transformations. Comm. Math. Phys. 28, (1972), p. 259. E. Cunningham, The Principle of Relativity in Electrodynamics and an Extension Thereof. Proc. Lond. r~ath. Soc. 8 (1910), p.77, and H. Bateman, The Transformation of the Electrodynamical Equations. ibid., p. 464. T. Fulton, F. Rohrlich and L. Witten, Conformal Invariance in Physics. Rev. of Modern Phys. 34 (1962), p. 442. J.A. Lester, Cone Preserving Mappings for Quadratic Cones over Arbitrary Fields. Can. J. Math. 29 (1977), p. 1247.
574
5. 6.
l.A. Lester
J.A. Lester, ConformaZ Spaces, J. of Geomtry 14, (1980), p. 108. E. Snapper and R.J. Troyer, Metpic Affine Geometry, (Academic Press, 1971).
Mathematiches Seminar der Universitat Hamburg Bundesstr. 55 2000 Hamburg 13 Federal Republic of Germany Present address: Dept. of Pure Mathematics University of Waterloo Waterloo, Ontario Canada
Annals of Discrete Mathematics 18 (1983) 575-580 © North-Holland Publishing Company
575
THE VAN DER WAERDEN CONJECTURE J.H. van Lint
1. INTRODUCTION For the world of combinatorics the 1980's started off quite nicely with the appearance of G.P. Egoritsjev's elegant and elementary proof of the famous Van der Waerden conjecture (cf. [2], [9]).
In the meantime at least two expositions of
the proof have been written (cf. [4], [6]) but at the time of the Rome meeting these were not available yet. Therefore it seemed appropriate to show a large gathering of combinatorialists how the proof works. If A is an n
x
n matrix with entries a .. (i lJ
l, •••• n; j
l •••.• n) then
the permanent of A, which we denote by per A, is per A : = ~ ala(l) a2a (2) ... ana(n) •
(1.1)
where a runs through all permutations of 1,2 •••• ,n. co 1umns of A as vectors al , •.. ,a in lR -n
n
We usually consider the
and we therefore wri te
per A = per (a- l , -a .... , -n a ) • 2 where
~j
=
(a lj , a 2j , ... , anj)T • (j
=
1, ... , n) •
It is well known that the permanent is a linear function of a. (for each j). -J
In
fact we have per A
(1.2)
a .. per A (i Jj) • (for any j) • lJ
where A(iJj) denotes the matrix obtained from A by deleting row i and column j. Permanents play an important role in several counting problems in combinatorics.
We mention the probleme des rencontres, the probleme des menages,
systems of district representatives. l-factors of graphs, the dimer problem and we give one geometric example.
Consider all 1-1 mappings (not correlations only) of
the points of a projective plane onto the lines such that each point is incident
576
I.H. van Lint
with its image.
How many are there?
The answer is the permanent of the incidence
matrix of the plane. In design theory (O,l)-matrices with constant line sum play an important role.
After dividing by this line sum we obtain matrices which belong to the
class of so-called doubZy stochastic matrices of order n, which we denote by nn' A matrix A is called doubly stochastic if all entries are non-negative and all The set r. is a convex polyhedron with the permutation matrices
line sums are 1. as vertices.
n
The simplest matrix in the interior of n is the matrix J which has -1
n
all entries equal to n •
n
The l'an der Waerden conjecture states: ( 1.3)
If A E n and A f. J then per A > per J • n n n A matrix A =
will be called a minimizing matrix if per A= min{per SISEn }. n n There is extensive literature on the Van der Waerden conjecture for which we refer II
the reader to the book Permanents by H. Minc [7].
We mention the most important
partial results. (1.4) If A E n is a minimizing matrix and a .. > 0 then per A(i Ij) n lJ
= per A.
(1.5) If A E n is a minimizing matrix then per A(ilj) ;;;. per A for aU i and j. n
It was known that the following improvement of (1.4) and (1.5) would be sufficient to prove (1. 3) : (1.6) If A E
'h
is a minimizing matrix then per A(ilj) = perAfor aU i and j.
The proof of (1.6) given by Egoritsjev depends on an inequality of A.D.
Alexand~f
which we treat in the next section.
2. ALEXANDROFF'S INEQUALITY In 1937 A.D. A1exandroff [11 proved an inequality for so-called mixed volumes of compact convex sets (also called mixed discriminants). also been announced by W. Fenche1 [31. R. Schneider [81.
The theorem had
In 1966 the inequality was generalized by
It seems that for a long time the inequality was only known to
geometers working in convex geometry (cf. e.g. [5]).
Recently a number of
combinatoria1ists apparently independently realized that the inequality was also useful for them.
Below we shall give a direct proof of the inequality which
577
The van der Waerden conjecture
Egoritsjev derived as a special case of Alexandroff's inequality. This proof is taken from [61. THEOREM 2.1: Let 2.1,2.2"" '2.n-l be vectors in bE IRn. Then (2.2)
(per(2.1 ,2.2"" '2.n-l ,E))2
~
and equatity hotds if and onty if
n
R with positive coordinates and tet
per (2.1 , ••• '2.n-l '2.n-l) ·per(2.1'··· '2.n-2'~'~) ~
= A2.n_l
for some constant A.
REMARK 2.3: Clearly the inequality (2.2) is also true if we only require that the In that case the claim about the
coordinates of the vectors -1 a. are non-negative. consequence of equality cannot be made.
We shall prove Theorem 2.1 using the concept of a Lorentz space. n following we consider IR with the standard basis.
In the
DEFINITION 2.4: The space mn is called a Lorentz space if a symmetric inner product
~,t>
= ~TQr
has been defined such that Q has one positive eigenvalue and
n - 1 negative eigenvalues. We call a vector
~
and isotropic if
~,~
~,~
positive (resp. negative) if
~,~
is positive (resp. negative)
= O. By Sylvester's theorem there is no plane such that
is positive on this plane
(~
; 0).
The following lemma is a consequence of
this fact. LEMMA 2.5: If 2. is a positive vector in a Lorentz space and
~..!?2 ~ ~f>. ~,E.> PROOF: If
~
and equality hotds iff
=
is arbitrary, then
A2. for some contant A.
is not a multiple of 2. then the plane spanned by 2. and
isotropic vector and a negative vector. form in A.
~
~
Consider
~
+ A~.
~
contains an
2. + A..!? as a quadratic
Since this form is 0 resp. negative for suitable values of A it has a
positive discriminant. 0 Consider vectors 2.1.2.2 .... '2.n-2 in mn with positive coordinates. Let!i denote the ith basis vector (1 ~ i ~ n). We define an inner product on IR by
(2.6)
.- per (a-1.•-a 2•••••-a f I2.x.r). --
578
J.R. van Lint
i.e. where Q is given by q1'J' .- per (a- l ,a- 2 ,···,a-n- 2,e.,e.) -1-J
(2.7)
per A(i,j In - l,n}, where A is a matrix with columns -al, ... ,a. -n a 1 and a • -n-fI
Note that at this point we do not use
THEOREM 2.8: IRn with the inner product given by (2.6) is a Lorentz space. For n = 2 we have Q = (~ ~) and the assertion is true. l Now assume that the asserti on is true for IRn- • We shall fi rs t show that Q does
PROOF: We use induction.
not have the eigenvalue O.
(2.9)
Suppose Q£ = Q.
per (a- 1 .a- 2, •.. ,a-fI- 2,c,e.) = 0 for 1 --J
Then by (1.2) ~ j ~
n.
Consider the n - 1 by n - 1 matrices
and apply the induction hypothesis and Lemma 2.5.
o = per
(al,· .. ,a-n- 3,a-n- 2,c.e.}2 - -J
~
We find
e .) per (a- 1,···,a-fI- 3,a-n- 2,a-n- 2.-J per (a- 1,···.a-n- 3' -c.c.e.) --J •
Since the first factor on the right-hand side is positive it follows that for 1
1, and let Z(a) ~ z(a). Let p = z(a)-l.
By Lemma 1,
there exist p transpositions Tl ,T 2 , •.• ,T p such that Ti connectsa T2••• T _ and 1 i l aT 1T2 ••• Ti _l • Thus, z(aT 1T2••• Tp) = 1 and z(aT 1T2••. Tp) a Z(a)-p = z(a)-z(a)+l. Now if a and a have the same parity, then z(a)-z(a) is even, z(a)-z(a)+l is odd and we are in the first case considered above, with aT 1T2 ••• Tp playing the role of a and aT1Tc ••• T that of a. If a and a have different parity, the we are in the p second case, q.e.d. THEOREr4 2: Let a,a
E
Sn and Zet
M = max {z(aS) + z(aS) }. S E Sn Then
-1
MEln+z(a a). Let S be such that
PROOF:
z(aS) + z(aS) If as
M.
= identity, then S = a -1 and M has the stated value. If as # identity, let
T be a transposition disconnecting as. S'
= ST
we would have Z(aS') + z(aS')
Now T cannot disconnect as, otherwise with
= M+2,
contrary to the choice of S.
Thus
there exist transpositions T ,T , .•• such that Ti disconnects aT 1T2••• Ti _l and l 2 connects aT T ••• T _ ·, and such that at each step the value of the sum is M. The 12 i l
584
A. Machi and P. Pizzi
process ends when, for some k,
~T1T2 •••
Tk = identity.
-1
Then S = 'lT 2··• k
= a
-1
and M = n+z(a a), q.e.d. Note that the integers m and Mgiven by the theorems above have the same parity (which is even if a and
a
have the same parity and odd in the other case)
and that this is also the parity of any other integer i such that i=z(aS)+z(aS) for p
E
Sn.
We now consider problem (iii) of the Introduction in the case of a
permutation S E Sn not necessarily circular. THEOREM 3: Let ~,a
E
Sn, m and Mas in Theopems 2 and 3, and Zet
be an integep
such that
m~ i
~
Mand i
=m (mod
2).
Jhen thepe exists S E Sn such that Z(~B)
+ z(aS) = i.
PROOF: It is sufficient to prove that given j > m and S E Sn such that z(aS)
+
z(aS) = 1, then there exists S'
Then starting with z(~S)
j
E
Sn such that z(~S') + z(aS')
= Mwe get the result.
j-2.
Assume
+ z(aS) = j.
If z(a0) and z(aS) are both greater than 1, then Lemma
applies, and if , is a
transposition connecting as and as, then S' = S, gives a sum equal to j-2. sume Z(aB) = 1.
Then z(aS) > 2, otherwise j = m.
and a611 each have at least two cycles. as'l'
Let '1 connect as.
As-
Then as'l
By Lemma 1 again, let 12 connect as'l and
Now with 6' = S1 11 2 we have a sum equal to j-2, q.e.d.
REMARK 1: Note that throughout this section we have not required that the group generated by
a
and
a
be transitive.
3. MINIMUM AND MAXIMUM VALUE OVER THE SET OF CIRCULAR PERMUTATIONS In this section we consider the three problems as stated in the i.e. we restrict our attention to circular permutations. THEOREM 4: Let a,a
E
Sn and let
Introductio~
Results about the genus of a pair of permutations
m = min ~ E
{Z(a~)
+
z(a~)
585
}
C
where C is the set of oiroular permutations of Sn. (i)
If a and a have different parity, then m = 3;
(ii)
If a and a are both odd, then m = 4;
Then we have:
(iii) If a and a are both even, then m = 2 or m = 4. PROOF: (i).
By Theorem 1. there exists S E Sn such that
z(aS) + z(aS)
= 3.
Assume z(aS) = 1 and z(aS)
a
2.
If S is circular we are done.
Otherwise let Tl
connect S and as (Lemma 1). Then
z(S) - 1 transpositions T1.T 2' •••• Tp such that Ti connects ST T2.•• T _ and only one of aST T ••• T _ and aST T ••• T _ • At the end of the 1 1 2 1 2 i 1 i l i 1 process we have ~ = ST 1T2.•• Tp' a circular permutation. and
Thus we can find p
z(a~)
+ z(a~)
= 3.
(ii) By Theorem 1. there exists S E Sn such that z(aS) + z(aS)
= 2.
Note that S cannot be circular. and as a matter of fact S must have an even number of cycles.
Thus if a is odd and
~
is circular. then
Starting from S we now construct a circular permutation connect S.
~
Z(a~) ~
2 so that m ~ 4.
such that m = 4.
Let Tl
Then:
Let T2 be a transposition connecting ST 1 and aT l (Lemma 1). Then either T2 connects aT 1 or it disconnects it. so that either
or
Note that ST1T2 cannot be circular. since p
= z(S)-l must be odd. Thus. a trans-
586
A. Machi and P. Pizzi
position '3 connecting ST T either disconnects the first two permutations (in the 1 2 first case) or it can be chosen as to connect ST 1'2 and 0'1'2 in the second case, In either case, the sum Z(aS") + Z(OS") = 4, where S" = s'lT 2'3' We can therefore choose p = Z(8)-1 transpositions 'l"2"""p such that, for even, either z(aB'1'2" "i)= 1,
z(oS'11 2,· "i) = 1,
Z(81 1'2"' "i)
z(B)-i,
or one of aB'1'2"'Ti and 081112""i is circular and the other one has three cycles,
Let now i
circular and
z(a~)
+
p-l.
In the first case, 'p is such that 811'2"'1p
z(o~)
= 4,
~
is
If we are in the second case, then 1p can be
chosen as to connect B'1'2 •• , l p_l and the non circular one of the other two, In any case, ~ = 611'2""P gives rise to a sum equal to 4. 2 must have an odd (iii) In this case a permutation B such that z(aB) + z(08) number of cycles, and an argument similar to the above shows that either m = 2 or m = 4, q.e.d.
REMARK 2: It is easily seen that there are pairs of even permutation a,o such that m = 2 and others such that m = 4. and
0
= (1,3)(2,4): here m
o = (1,2,3)(4), then m = 2. lem:
unde~
For instance, for n = 4, let a = (1,2)(3,4)
4; if instead we take a = (1,2)(3,4) and The above theorem leaves it open the following prob-
What conditions on a and 0 is the minimum of the sum
aU ail'aulal' permutations ~ of Sn equal to 2?
z(as)+z(o~)
over
Note that the examples gi ven above
show that the transitivity of the group generated by a and 0 is not relevant. COROLLARY: Let a,o
E
Sn and Zet the group generated by a and 0 be transitive.
Then the maximum value
~
of the sum
g(a,~)
eiraular permutations of Sn is given by: (i )
if a and
0
have different parity, -1
j.J
(ii)
-if a and ~
(i i i)
= g(a,o) 0
n+z(a 0)-3 are both odd, -1
a 9 (a,o ) +
11 a
and
0
2
+
n+z(a 0)-4 2
are both even, then either
w = g(a,o) + n+z(a
-1
2
0)-4
+
g(o,~)
as
~
varies in the set of
Results about the genus of a pair ofpennutations
g(a.a) + n+z(a
-1
2
a)-2
Let now< a.a> be transitive. -1
exceed n + z(a a).
587
By (2) and (3) the sum Z(az;) + z(az;) cannot
That there exists
z;
such that this value is actually attain-
ed is shown in [2].
We sketch the proof here. Asis transitive. so is If a is circular, then z; = a- l will do. Otherwise. by the transitivity
.
of there exist p transpositions '1.'2 .... '" .
-1
P = z(a)-l such that ,. p. 1
connects a'1'2""i-l and dlsconnects a a'1'2""i-l' Then, lf z; = a'1'2""p' z;' =z;-l is the required circular permutation. We state this in the form of a theorem. THEOREM 5: Let a,a
E
Sn such that is transitive, and let
M= max {Z(az;) + z(az;) } z; E C where C is the set of circular pe~utations of Sn.
Then
-1
M=n+z(a a).
Let now m and M have the meaning of above. arises as to whether given an integer i. m ~ i
As in section 2, the question
~
M. having the same parity of m
there exists a circular permutation z; such that the sum z(az;) + z(az;) is precisely i.
The answer is given in the following theorem.
THEOREM 6: Let a,a
E
Sn, a F a.
Let
m = min {Z(az;) + z(az;) } z; E
C
and
M= max {z(az;) + z(az;)} z; E C where C is the set of circular
m~ i
~
Mand i
Then there exists z;
E
=m (mod
pe~utations
2).
C such that
z(az;) + z(az;) = i.
n
of S.
Let
be an integer such that
588
A. Macht and P. Pizzi
For the proof of this theorem we need two lemmas.
LEMMA 3: least three
Let a and
0
~lasses
and
~o distin~t
be
at least two.
0
Then there exist
the set, belonging to different alasses of both a and the w,,:on of the alasses of i and j in a and (ij) su~h
.'nent k of the set
k E (ij)
0
~o
that in
o
elements i and j of
su~h
and
0,
that a has at
that if (ij)
a
is
there exists an de-
that
(ij) \ (ij) n (ij) •
U
a
su~h
partitions of a set
a
0
0
PROOF: Let i and j be as in Lemma 1.
Assume first that (i)
a
= (i) 0 and
(j) a (j) 0 (where the notation is self-explaining). As a f 0, there exists £ such that (£) f (£). Let k E (l) \ (,I.) (or k E (.~) \ (£». Then i and £ are a
a
0
0
k ffi (i) ( U1(i) 0
(j)
U
a
(j)
U
0
and t
E
t rf: (i)
o
U
(j)
(k) ,
U 0
0
LEMMA 4: Let a,o
then
Sn, a f
E
partitions of {1 ,2, ... ,n} ~o
have at least
z(a~) + z(o~)
If t
E
(k)
E
(i) 0\ (i) a ).
If t
E
(j)
o
,then j, k and t will do.
o
t and i will do, q. e. d.
j,
0, and let indu~ed
classes.
Assume now (i) a f (i) 0 ,
(i) a\ (i) 0 (or t
just take j and k as in Lemma 1 and t.
a
0
as in Lemma 1 and k is the element we seek.
~ be a ~ir~ular permutation su~h that the
by the cyales of
a~
and
o~
are the same and
Let
= i.
Then there exists a circular permutation r;' such that
z (a~') +
Z (0 ~ ') =
i
and such that the partitions of {1,2, ••• ,n} induced by the cycles of ar;' and or;' are different.
PROOF: Since that
al;
a f
0,
we have
takes k into land
a~ 01;
f
Then there exist two elements k and £ such
01;.
takes k into t, where t f £.
not belonging to the cycle of k and
l
(11;
(l, .•. ,s)(k,l, .•• ) •••
01;
(l, ... ,w)(k,t, ... ,l, ... ) ...
and we have either:
in
a1;
and
01;.
Then:
Let 1 be an element
If
Results about the genus of a pair of permutations ~
(l ••••• u.k.v ••••• I •••. )
~ =
(l •.••• u.l.v ••••• k•..• )
589
or
(where u and v can be equal to l.k or I).
If
~
is of the first type. then by
multiplying by (l.k.!) we have a~(l.k.l)
(l ••••• s.k)(I ••.• ) •.•
a~(l.k.l)
(l, •••• w,k.t •••• )(!, ••• ) ...
~(l
,k.l)
I,; I .
Now the cycle of k in t.
Moreover. t
F k,
al,;
contains t. whereas the cycle of k in al,; does not contain
since if no such t between k and I exists in
would take k into I, contrary to assumption. z(a~') + z(a~')
and the cycles of
a~'
The case in which
1,;
al,;.
then
as
Then we have:
=i and al,;' do not induce the same partition of {1.2, •••• n}.
is of the second type is dealt with in a similar way (multiply
by (l.!.k)). q.e.d. PROOF OF THEOREM 6: It is sufficient to prove that if z(al,;) there exists a circular permutation starting from a circular permutation both equal to 1. al,;
~'
such that
Z(a~')
1,;
such that
Z(a~) + z(a~)
Thus assume i > m.
we get the result.
+
+
This implies that
z(al,;) = i > m. then
z(a~')
z(a~)
= i-2. Then
= M. the maximum. and
z(a~)
are not
By Lemma 4 we can assume that the partitions induced by as and
on n.2 ..... n} are different.
(i) al,; has at least three cycles and al,; at least two. i.j and k such that: a~
(i, ... )(j, ... )(k .... ) ...
a I,;
(i ..... k .... )(j .... ) ...
Then if
i
l
is the transposition (ij) we have
al,;i 1
(l .... ,j .... )(k .... ) ...
a~il
(i ..... k..... j .... ) ...
I,;i
l
(i .... )(j .... ).
Then by Lemma 3 there exist
590
A. Machi and P. Pizzi
If k belongs to the cycle of i of 1;'1' then (j,k) connects the first and the third permutation and disconnects the second one. ~Tl'
If k belongs to the cycle of
j
of
Then with '1 = (i,j) and '2 = (j,k) or '2 = (i,k~
the same happens for (i,k).
s' = ~'1'2 is the required circular permutation. (ii) i
z(a~)
= 1. If
Z(a~) ~
3, the either i
= m+2 and in the latter case z(a~)
Assume ing
G~T1
= m, contrary to assumption, or
is the circular permutation giving the minimum.
~'
> 3 and let '1 be a transposition connecting aT and '2 one connect-
and
~~l;
if such a '2 can be found as to connect
the required circular permutation.
~1;'1'
then
~' = ~'1'2
is
Otherwise,
and we are in case (i). (iii) z(al;) = 2.
Then we can assume
REMARK 3: If a = 0, then given sible to find
~'
~
z(o~) ~
such that the sum equals i it is clearly not pos-
such that the sum equals i-2.
s' such that the sum equals i-4 (for i COROLlIl.RY: Let ",a m=
min ~ E
3 and we are in case (i), q.e.d.
E
Sn, a "!
{g(a,~)
° and
~
It is possible, however, to find
m+4, of course).
Let be transitive.
Let
+ g(o,l;) }
C
and \J
=
max {g(a,r;) +
1; E
g(o,~)
}
C
;,;here C is the set of eil'euLar pel"T7lutations of Sn. that m .;; i .;; 9(~,1;)
fl,
then thei'e exists
+ g(a,l;)
I; E
Then if i is an integer suah
C sueh that
= i.
BIBLIOGRAPHY 1. 2.
A. Jacques, Sur le genre d'une paire de substitutions, C.R. Aead. Sai. Paris, 267 (1968). R. Cor; and A. Machl. Su alcune proprieta del genere di una coppia di permutazioni, Bollettino U.M.l., (5) 18-A (1981). 84-89. Istituto Matematico "G. Castelnuovo" Citta Universitaria 00185 Roma Italy
591
Annals of Discrete Mathematics 18 (1983) 591-.
{l',si) ~ if all lines
X E L \ L contain at least th1"ee points, then parallel affine lines have the same
a
::'.7.p'_{i:nc
l i tu.
Let us now prove the theorem:
= (P,L,L a ,1/ )
THEOREM 1: Let L
affine i<el'ne 1s ape at least 2-dimensional;
:;.) ti;:::
; \ E.b
be a regula1" exchange p.p. space such that:
and Vl ' are
f~J..Zfi.lled;
,-'wn thr:: affine ke1"nels a1"e desarguesian affine spaces.
The proof is given by the following lemmas.
2.4
~:ndel"
the asswnptions of Theo1"em 1
the Itaffine Veblen axiom" V2 holds
in the fol"ms: ~ i) - Let R, S E La' with R n S "I/J and S
E S \
Rand pER, we have (p II b"s}) n S
by 1.1 iv), R () S
* I/J'"
{1",s}
E
IRI ;;;.
* I/J
3.
Then for r
(Tmpezaxiom in [51) (note that,
La'
i i) Let R,S E La' with R n S '" I/J, 1" E R \ Sand s E S \ R.
(Y' II S) () (sll R)
* I/J
R \ S,
E
Then we have
(Parallelog1"amm=iom in [5]) (If affine lines have at least
three points, then ii) follows from i) (see [51); vice versa ii} is indipendent from i) if affine lines have at most two points).
PROOF: Let {v}:= R n S, let V be a projective line through v (existing by let;.; to V \ {v}.
Let us project the trian!]le (v,r,s) from a pOint q
E
~)
and
V \ {v,w},
Partial parallelism spaces and slit spaces
existing by 2.1, onto <w>.
595
Since VEL ,<w> '" and moreover q f/:. <w> p
Let R':= (w#R) and S':= (wIIS).
By 2.2, Vl'
is equivalent to
~
and, by
.
U
~,
with
respect to the lines {q,r}, {q,v} and {q,s}, {q,v}, there exist points {r'} .R' n {q,r} and {s'}:= S' n {q,s}.
Since {r,s}C =, {r',s'}C = and k'" E=. Therefore k
E
(sll R) n (I'll S).
2.5 Under the assumptions of Theorem 1, and if att tines X E L \ La contain at teast three points, att affine tines have the same cardinatity n
PROOF: By
~,
~
2.
all affine lines through the same point have the same cardinality;
hence, by 2.3, we obtain that all affine lines have the same cardinality n
~
2.
2.6 An affine paraUet structure which futfns the affine Vebten axiom is an affine space.
(For the proof, see Lenz [5] ). PROOF OF THEOREM 1: By l.lii), the affine kernels are affine parallel structures in which, by 2.4, the affine Veblen axiom
~
holds.
Then, by 2.6, the affine
kernels are affine spaces which are desarguesian if their dimension is at least three, or even if their dimension is two and If R E La IRI = 2.
Now, let us assume
that there exist affine lines containing at least three points: in order to complete the proof of the theorem, we have only to consider the case that the dimension of affine kernels is exactly 2. If a,b,c, a',b',c'
We have to show that If v E P and
E , with {a,b} II {a',b'}, {b,c} II {b',c'}, {a,e}/I {a',c'},
{v} = {a,a'} n {b,b'} , the inclusion v E {c,c'} holds.
Let V be a projective line through v; now let us project, as we made in 2.4,
596
M. Marchi and S. Pianta
the triangles (v,a,c), (v,c,b) and (v,a,b) from a point q oJ E
E
V \ {v} onto< w>, with
V \ {v,q} (existing by 2.1). Applying y..! to the lines {q,a}, {q,b} , {q,c} respectively and to {q,v} we
obtain the following points: {a"} : = {q,a} n (wll {v,a}) , {b"} : = {q,b} n n (vii {v,v J) , {a"; : = {q,c}n (w/! {v,c}). lq,c}
Applying now Vl to the 1ines {q,b} ,
resp. and to {q,a} , we have: (aliiI {a,b}) n {q,b} f. 4>, (a"/! {a,c}) n
n {q,c}
f
4>; hence by {q,b} , {q,c}
rf. <w>we obtain
{a",b"}/! {a,b}
and
Again by Vl applied to the lines {q,b} and {q,e} we have
{al,e"}11 {a,e}.
(IJ· II {b ,c)) n {q ,e}
* 4>
and then, because {q ,e} rf. <w> , we have {b" ,e" }II {b ,e}.
In the linear variety V U {a'} , the line {a",a'} intersects the line V in a point :; different from v, w, q. Then applying Vl to the lines {z,a"}. {z,v} resp. and {a' ,b'}
n {z,b"}
applying now n
f.
{
and b'
E {z,b"}
{~}
we obtain
because {z,b"} rf. ;
to the lines {z,a"}, {z,b"} resp. and {z,e"} we obtain {a',e'} n
¢, {b' ,e' }
n {z,e"} '"
¢
and since {z,e"} rf. , e'E
{z,e"};
applying
to the lines {z,e"} and {z,v} we obtain {v,e} n {z,e"} F 4>; then since
~
{v,e}n {z,e"} =
n {z,e"} = {e'} we have e'
E
{V:;}, that is
v E {e,e'}.
3. EMBEDDING OF REGULAR EXCHANGE P.P. SPACES INTO PROJECTIVE SPACES In [3] the so called slit spaces are defined: they are projective spaces from which a projective subspace, or a line, or a point, has been deleted. Let us consider exchange p.p. spaces
(P,L,L ,/!)
L
a
which are regular and
fulfil the following axiom: ~
L n M
- Let L,M
* 4>
(l U {p}) n
3.1
with dim (L U M)
=
2; it holds
3 pEP \ L U t~ such that (L U {p}) n (-M-U-{-p-)) E L. (t~
U
{p })
V2 and
- II R,S
~
E La
{r ,s]} n {r ,s2} 1 2
*¢
(Then we have:
n L U M E P).
In a regular exchange
at Least 2, ~
L\ La
E
imply
p.p. space L whose affine kernels have dimension
the axiom:
with RII S, II r ,r E R, II l 2 or {r ,51} II {r ,8 }. l 2 2
5
,5
1 2
E
S, it holds:
PROOF: Let = ; then by 2.6is an affine space where V3 is always 1 fulfi lled II R,S C < r] >.
Partial parallelism spaces and slit spaces
*
*
597
*
Suppose now and Pl P 2 , sl s2; let p be any point of \ Rand Hi := (si# {Pi,p}) (i=l,2). Since Pl 1'2 we have Hl HH2• If every affine
*
kernel has dimension 2, then it follows that there exists {q} : = Hl II H • If 2 the affine kernels have a dimension, greater than 2, then we Rave to show that Hl and H2 lie in a linear variety of dimension 2, hence by V2 and 2.6, in an affine plane. Let us consider the linear variety r : = R U {P,sl}; then dim (r)
*~
*
*
and hence r n r (because (i=l,2) and Hl
r n = R U {p}
dim (r II ;
R \ A and Ii a l ,a 2 E A \ R : {l"l,a l } n {1"2,a 2 } '" rf>. By ~, --with respect to the lines R = {1"1 ,1"2} and {1"2,a }, we obtain {a' }:= (1"11 A) n show that Ii i'1'1"2
E
2
n {1"2'u 2 }. ~, with respect to the lines {1"1 ,a' } II A gives {1"1 ,a l }l{r 2,a } f ¢. 2 Thus the Veblen axiom has been proved also for any two intersecting lines of 11, one of them being affine. iii) Since
L
is an exchange space, we can always assume that
by two lines R,$ as in case i).
11
is generated
By i), ii) any two lines R',$' intersecting Rand
$ in distinct points have a common point or are parallel and, moreover, they both intersect the affine line A or one of them is parallel to A and the other one intersects A. iv) For the lines R',$', intersecting R,$ in distinct points, we can now Prove that
s'} n {;;-;o} '" or l' 1'" 2 E R' \ $' and s'l' S'2 E $' \ R' ' {p'1'1 2' 2 'f' {l"l,sl} II {1"2,s2}' because for R' ,$' it is possible to repeat the arguments used I.J
1'"
for Rand $ in i) or ii).
A.
I.J T
y
Then any line intersecting R' ,$' in distinct points is
also intersecting Rand $ and, if it is non affine, by iii) it intersects the line A too. In this way, all lines of R n $ with points of A.
11
either intersect R,$ in distinct points or join
With the arguments i) or ii) all these lines are inter-
secting in one point or are parallel to A and this means that
11
is a pointed
plane. 3.3 In a regular exahange p.p. spaae r, Vl, V3 and
~
imply that the 2-di-
Partial parallelism spaces and slit spaces
mensionaL Linear varieties
TI
599
of E not containing affine Lines are projective
planes.
PROOF: Let v, r, s be any three distinct points such that show that II rl'r 2 E R:= {v,d and II sl,s2 n {r 2 ,s2} '" ~.
E
TI
= {v,r,s}.
We have to
s:= {~} it holds: {rl,sl} n
Let us denote for short Tl := {rl,sl} , T2 := {Y'2,s2}' affine line through v and p := R u A, a := SuA and
r~oreover
let A be an
= R u S; p and a are 2-di-
TI
mensional linear varieties with an affine line and a non affine one, so that they are not contained in any affine kernel.
By 1.2, p and a have exactly one
parallelism class of affine lines and then, by 3.2, they are pointed planes. q
be any point of
a\
Let
(A u S) and let us project the lines T1 and T2 from q onto
by means of the linear varieties 'i :=
{q}
p
u Ti (i=l,2).
In order to obtain the projections 'i n p (i=l,2) of T1 and T2 onto p, by and the definition of linear variety, we have to consider the intersections
~, ~,
of the lines {q,r i } and {q,si} (i=1,2) with p. Since r E p, {q,r }np= {r } i i i (i=l ,2). Moreover since A = p n a and sl ,s2,q E a, we have {q,si } n p = = {q,s. } n A. 1
Now, two cases are possible.
a) {q,si} are non affine for i=l,2; then by 3.2, since {~f {q,si} n A a : {hi} for i=l ,2.
' A C a,
Hence {hi,Y'i} = 'i n p (i=1,2) are non affine
lines. b) One of the lines {q,sf is affine; let {q,sl } be an affine line, then, by 1.2ii), {q,s2} is non affine.
Thus '1 n p is the affine line (1'1# A) because
{q,sl}11 A, and '2 n p is a non affine line.
At any rate, '1 n p and '2 n p are two lines of
p
which is a pointed plane,
so that these two lines (not both affine) must intersect, by 3.2, in a point = ('1 n p) n ('2 n pl· that the lemma is proved. {t
}:
Then '1 n'2 = {q,t} and hence,
by~,
T] n T2
*~,
By 2.4,3.1.3.2.3.3, we observe that in a p.p. space E=(P,L.La ,#) which fulfils the assumptions of Theorem 2 it holds: L = La U Lp , and moreover L is regular in the sense of Karze1 [3], [4].
Hence
L
is exactly a slit spaoe (P,L).
For any slit space it holds (see [3]. [4]): 3.4 Let (P,L) be a slit space of dimension n.
Then theY'e exists a unique
so
600
M. Marchi and S. Pianta
n-dimensional- projective space II = (P, L) and a pro.iective subspace V* V such that: i) p = P \ V , U) L = {L \ V
Lp
{l
E
By
L: [
3.4,
n V =
I
L I/>}
E
L, [~ V},
and La = {[ \ V
I[
LE L,
n
vi
is then imbedded into the projective space
3.4 we obtain that, for any a
E
P, the set
U
V
l}.
and by the proof of
II
is a projective subspace of
projective closure od the affine space ; hence dim (V)
=
dim (J(c),l) •
E' exactly in the point
PROOF: (i), (ii)- By 2.2, the mapping i is the required isomorphism. [a,l1l' Bc " {[ a+\l(ch.lf;(C),\lE;1 :
E;
(iii) Since
E M} we have
[a-c+\l(c),ll ,,[ a,l1]' Bc () pl. 3.2 Let /\
oe
a kinematic subspace of
BC E L(f 0,11 ),
1;;1. th
Bc
E
=( P, L,· )with Aif A and I\' : = 1\ () A.
For any
we have /\ "I\' . Bc •
C /\,
PROOF: Every line of the partition {(p
II~
Bc) : p E /\} of/\ meets A' exactly in
one point, by 3.1, and so {(p
lit Bc) :
Since (a II £Bc' 1\ " U{ (a
=
p E I\}
= {(a
/1~Bc):
a
E I\'}.
[a, 11 • Bc we have
I I 'l.B c) : a
E/\')
= 1\'.
Bc'
Now we investigate if, in the kinematic space E introduced by Theorem 2, there are
-
-
lines L ELand points pEP \ L such that: (p
119. L) "
(p
Ilr L).
609
On a class of kinematic spaces
The following properties hold:
3.3 For any BEL ([ 0,1] ) and any [b, v] ¢ B one has
a
a
PROOF: By 1.3 and 2.8, we have: ([ b,v] II, Ba) = [b,v] • B = Bb ()[ b,v] and ([ b,v] II B) " a +v a r a
B • [b,vj. a
This implies: [b,v] . B = B .[ b,v] a a
3.4 For any X E L(O) ~
b+v(a)=a, that is b=a-v(a) and [b,v] = [a-v(a) ,v] E B . a
L([ 0,1]
) and any [b,v] ¢ X one has:
([b,v] II ,Q, X) = ([b,v] II r X)
v(X) = X.
PROOF: It is evident by 2.9.
4.
KINEr~ATIC
EXCHANGE SPACES
Let L = (P,L,.) be a kinematic space (cf. Definition 1.1) and L' =(P',L' ,.) a kinematic subspace of L.
For any pEP, the point subset P:= p.p' is called
proper tinea!' variety of L.
By a tinea!' variety we understand a proper linear
variety, the empty set and all the points of P. 4.1 For any a,b
E
PROOF: Let a' :=p
-1
therefore a,b ~ P.
P:= p.p', ·a, b':=p
a:tl ~ P. -1
-
--
·b E p', then, by axiom 1.3, a,b = p.(a' ,b') and
~
~
4.2 For any proper tinea!' variety P:= P'P', there exists a kinematic subspace
L"=(P",L",') such that P = P"·p. PROOF: With L'=(P',L',') is also L":=(P",L",·) L":={pXp
-1
\~ith
P":=P'P"p
-1
: X E L'} a kinematic subspace and we have P = P".p.
and
610
M. Marchi and E. Zizioli ~
4.3 POr' ani! a E P .- p·p'=P"·p and L ~ P, ;,,'ith L E L, we have
I I ~ L)
(a
~ 1-'
(allrL)CP.
and
PROOF: Let L = pL' = L"p and a=pa'=a"p with L'
~
P', a' E P' and L"
Since P' and pOI are kinematic subs paces we have: (a' II (a" I I
r
L")
C
-
P".
L')
~
C
-
~
p lI , a" E PII.
P' and
This gives us: ~
(a 11 9" L)
pta' II
(all r
(a"ll
L)
r
t
L')
L")p
p'P'
C
C
-
and
P ~
p
lI
p= P.
4.4 'The set-theor'etical inter'Gection of any set of lineal' var'ieties of
L 7:13
a
Unea:r ))ar'7:e ty.
PROOF: P = P • P' and
Q = q.Q' where P' and Q' are kinematic subspaces.
a E P n Q. then there exist pEP' and q ~ --1 ~ --1 have: P = ap . P' a· P' and Q =aq P n Q = a'P' n a.Q'
Q' such that a=
E
Q' = a • Q'.
P'P
Let
= q.q; hence we
Then
a(P' n Q').
Therefore P n Q is a linear variety because P' n Q' is a kinematic space.
By
the associative property of set-theoretical intersection one has the thesis.
DEFINITION 4.1: A linear variety P of a kinematic space CZC8Ul'e
if P is
of a point-set 1
containing I. set of P. min
~
-
~
{I I I
for any a.b E
4.5 For' I
the intersection of all linear varieties
-
As the dimension of a linear variety P we define the cardinal number: ~
I~P.I=P}-l.
if the exchange axiom
b
= (P,L.·) is called the
We shall write P:= 1 and the set I will be called a generating sub-
DEFINITION 4.2: A kinematic space
~
l:
E
{a} U I.
C
~
l:
= (P,L,·) is called kinematic exchange space
holds with respect to this closure operation:
P and for any subset I b
~
I
~
I.
a E {b} U
P and pEP, we have p.!
~
=
p·r.
P:
On a class of kinematic spaces
611
PROOF: The linear variety P'I contains the set p'I, hence p7I ~ p·I. Since -1 -1 --1 I =P p·I is contained in the linear variety p • p'I we have I ~ P • p·I hence p'I
4.6 E
~
p·I.
= (P,L,')
ex~hange
is a kinematic exchange space if and only if the
axiom
holds for the lattice y of aZl kinematic subspaces of E.
PROOF:
For I
C
P let be the smallest kinematic subspace of
If 1 E r then < I > =
E
containing I.
I.
First we assume that y is an exchange lattice. Let I ~ P with F~, -1 -1 -1 j E rand bE {a}U r \ I. Then j .b E j . ({a}u 1) \ j (1) = -1 -1 -1 -1 -1 -1 = {j . a } u j . 1\ j I by 4.5 and hence j . b E \ < j I>. -1 -1 -1 -1 From ~ we obtain j a E = j ·{b}u I -hence a E {b} u I. Now let
-::-r-
E
be a kinematic exchange space.
For Ie P and b E\ we have
b E < a u 1 u I> \ = a u 1 u I \
4.7 Let
~
TUI
and hence by ~: a E b u 1 u I =.
= (P,L,') be a kinematic exchange space of dimension n
exist a line L and a point q (q
~
~
3.
If there
L) such that:
= (q IlrL)
(q II~L)
then for all points pEP we have:
PROOF: Let M:=(qll L)=(qll L). r
~
For each point p
~
L,M we consider the linear
variety: P:= {'j)}UL n {p} u M. (i) For
p~
----= {p} U L would
{q}U L we have {p}U L., {p}U
imply p E {q} U L by
~;
dim "{j)TUL = dim {p} U M=2 by~.
{p} U L or {p} U 14 and dim
P
b.
This corollary is a simple asymptotic generalization of the result mentioned earlier, that no BCH code is quasi-perfect (b=l) except that with d=5. The Supercode Lemma.
If the code A is properly contained in the code B of
minimum distance d(B}, then cr(A} ;;. d(B}. COROllARY: The extended tr'ipZe-er'r'or'-oor'r'eoting BCH oode is not a pr'oper' subcode of any oode of minimum distanoe ;;. 7.
This simple and useful bound holds for nonlinear codes. in a special case in [6].
It led to the bounds (I) and (2).
It first appeared Generally it seems
harder to find useful upper bounds on covering radius than lower bounds. The redundancy bound.
The linear (n,k) code C satisfies cr(C} < n-k.
The Delsarte bound, [3].
The code C has covering radius at most its "ex-
ternal distance", which, when C is linear, is the number of different nonzero 1
weights taken by the vectors in the orthogonal, or dual, code C
.
621
A brief survey of covering radius
This result is difficult to prove.
The bound is exact for perfect codes and
for the m-odd case of the triple-error-correcting BCH code, and also for the poor Reed-Muller code RM(m-3,m).
But there is some slight evidence that the value of
this bound is well above the covering radius for most codes (see below).
It is,
however, hard to get general results on its value. The Pl + P bound, [14]. 2 check matrix
Let the (n,k) binary linear code A have parity-
I ·0 r'
where r=n - k is the redundancy of the code, and 0 is an rxk matrix of rank j .;;
min {r,k}.
(I r is the rxr identity matrix).
Let 0 be the parity-check matrix
of a (k,k-j) linear code of covering radius Pl , and let the covering radius of the (r,j) linear code consisting of the column space of 0 be P2· Then cr(A) .;; 01 + P2• This simple new result is better than the Delsarte bound for a majority of 14 mutually inequivalent (10,4,4) codes [ 14).
But that it bounds covering radius
in terms of covering radius may be a disadvantage.
It too is hard to evaluate
generally. Karpovsky
(11)
has recently studied t(n,k), the smallest covering radius for
any linear (n,k) code.
One simple observation is that t(n,k) .;; '!(n-k)i, where
Ixi denotes the least integer i > x.
This bound is satisfied by any code with Ir
and a column of all-l 's in the parity-check matrix.
Asymptotically it is the
dashed line in Figure 1, and it suggests that the sphere-covering lower bound on covering radius is a good one.
An improvement on this bound for high-rate codes
is the following result: If n,k,q, and m are positive integers satisfying n - qm> k ~ q(2m - m - 1) then t(n,k) .;; q +
I~(n-k-qm)i.
For details the reader may see
(11) •
622
H. F. Mattson Jr.. and J.R. Schatz
rate
1
""
"" "" ", cov~ng
packing
Elias
sphere
bound
bound
"
radiuSfn
Figure 1
ACKNOWLEDGEMENT: Partially supported by N.S.F. Grant MCS-7903409.
BIBLIOGRAPHY O. 1. 2. 3. 4. 5.
E.F. Assmus, Jr., and H.F. Mattson, Jr., Some 3-error-correcting BCH codes have covering radius 5, IEEE T~ans. Inform. Theo~y IT-22, (1976). 348-349. E.R. Berlekamp, Algeb~aic coding theo~y (McGraw-Hill, New York, 1968). E.R. Berlekamp and L.R. Welch, Weight distributions of the cosets of the (32,6) Reed-Muller code, IEEE T~ans. Info. Theo~y, 18 (1972), 203-207. P. Delsarte, Four fundamental parameters of a code and their combinatorial significance, Info and Cont~ol, 23 (1973), 407-438. f1.J.E. Golay, Notes on digital coding, P~oc. IEEE, 37 (1949). 657. N.J.E. Golay, Anent codes, priorities, patents, etc., P~oc. IEEE, 64 (1976), 572.
623
A brief survey of covering radius
6. 7. 8. 9. 10. 11. 12. 13. 14.
D.C. Gorenstein, W.W. Peterson, and N. Zierler, Two-error correcting BoseChaudhuri codes are quasi-perfect, Info. and Control., 3 (1960), 291294. R.W. Hamming, Error detecting and error correcting codes, Bell Syst. Tech. J., 29 (1950), 147-160. T. Helleseth, All binary three-error-correcting BCH codes of length 2m_l have covering radius 5, IEEE Trans. Inform. Theory IT-24 (1978), 257-258. T. He11eseth, No primitive binary t-error-correcting BCH code with t> 2 is quasi perfect, IEEE Trans. Inform. Theory IT-25 (1979), 361-362. J.A. van der Horst and T. Berger, Complete decoding of triple-errorcorrecting binary BCH codes, IEEE Trans. Inform. Theory, 22 (1976), 138-147. M. Karpovsky, H.F. Mattson, Jr., and J.R. Schatz, A survey of covering radius, to be submitted to IEEE Trans. Inform. Theory. T. Kasami, Weight distributions of Bose-Chaudhuri-Hocquenghem Codes, in: R.C. Bose and T.A. Dowling, eds., Combinatorial Math. and its Applications, (Univ. of North Carolina Press, Chapel Hill, NC, 1969) Ch. 20. F.J. Mac Williams and N.J.A. Sloane, The theory of Error-Correcting Codes, North Holland, Amsterdam, 1977. H.F. Mattson, Jr., An upper bound on covering radius, Proc. Int. Col log. Graph Theory and Combinatorics, Marseille, 1981; to appear in Annals of Discrete Math.
15. 16. 17. 18. 19. 20. 21. 22. 23.
I·'
23.
R.J. McEliece, Quadratic forms over finite fields and second-order ReedMuller codes, JPL Space Programs Summary, 37-58-111 (1969), 28-33. A.M. McLoughlin, The covering radius of the (m-3)rd order Reed-Muller codes and a lower bound on the (m-4)th order Reed-Muller codes, SIAM ,T. Appl. Math., 37 (1979), 419-422. J.J. Mikkeltveit, The covering radius of the (128,8) Reed-Muller code is 56, IEEE Trans. Inform. Theory IT-26 (1980), 359-362. W.W. Peterson and E.J. Weldon, Jr., Error-Correcting Codes, 2nd ed., (MIT Press, 1972). J.R. Schatz, On the coset leaders of Reed-Muller codes, Ph. D. dissertation, Syracuse University, Syracuse, N.Y., 1979. J.R. Schatz, The second-order Reed-Muller code of length 64 has covering radius 18, to appear in IEEE Trans. Inform. Theory IT-27 (1981). N.J.A. Sloane and E.R. Berlekamp, Weight enumerator for second-order ReedMuller codes, IEEE Trans. Inform. Theory, 16 (1970). 745-751. H.C.A. van Tilborg, Uniformly Packed Codes. Ph. D. thesis, Technical University Eindhoven, 1976. J. Wolfmann, Un Probleme d'extremum dans les espaces vectoriels binaires, Annals of Discrete Math., 9 (1980). 261-264. 15 N.J. Patterson and D.H. Wiedemann. "The covering radius of the~ ,16) ReedMuller code is at least 16276" IEEE Trans. Inform. Theory (to appear). School of Computer and Information Science Syracuse University Syracuse, New York 13210 U.S.A.
9427 Bullring Lane Columbia, MD 21046 U.S.A.
This Page Intentionally Left Blank
625
Annals of Discrete Mathematics 18 (1983) 625-634 © North-Holland Publishing Company
A GRAPHIC CHARACTERIZATION OF THE LINES OF AN AFFINE SPACE F. Mazzocca and D. Olanda *
ABSTRACT A set of axioms is given which characterizes the incidence structure (R(A),F(A)) where R(A) denotes the family of lines in an affine space A, and F(A) the planar pencils.
1. INTRODUCTION AND PRELIMINAIRES If P is a projective space coordinatized by a field, the Grassmann variety of the lines of P can be viewed as an incidence structure G(P)
=
(R(P),F(P)),
where R(P) is the set of lines of P, F(P) the family of planar pencils and incidence is by containment (the planar pencil F(TI,X) consists of all the lines in the plane TI through the pOint X).
This structure, which of course can be defined even
when P is not coordinatizable over a field, is called the Grassmann space of lines of P.
In a totally analogous manner, we define, for an affine space A, the inci-
dence structure (R(A),F(A)) where here R(A) denotes the set of lines of A and F(A) the family of affine planar pencils.
A pencil consists of all the lines in a
plane either through a fixed point (type 1) or in a parallel class (type 2). The spaces G(P) and G(A) are both partial line spaces and have been characterized in
[7J
and [11 respectively using properties of their maximal subspaces.
On the other hand, a characterization is given in [51 for the Grassmann space G(P) of lines of a finite projective space P based only on the incidence structure (R(P),F(P)).
Here, we extend this result to the case G(A) where A is any affine
space (finite or infinite). In the following S represents a non-empty set of points and R a covering of S by lines (so that every point is in a line). The pair (S,R) is termed a partial line space if two distinct lines have at most one point in common, and if every
line has at least two points. When two points P and Q share a common line we say
626
F. Mazzocca and D. OIanda
they are ecZli,waI' and write P - Q. write P f Q.
The line is denoted by (P,Q).
Otherwise we
For two lines rand s the symbol r - s means that each point of r is
collinear with each point of s.
The partial lines space is called
r~ope~
if it
contains at least two non collinear points. For every pair (r,P) with r E Rand P E S-R there are three cases to consider.
One says P and r are
and r are said to be P and rare one-aoL
8ke~,
Unea~,
eoplana~,
written P - r, if P - Q for every Q E r.
written P f r, if P f Q for all Q E r.
P
Finally, we say
denoted P ---! r, if there is a unique point Q in r col-
linear with P. A subset T is called a std!space if every pair of points of T is contained in a line consisting of points of T. In the following we will consider only proper partial line spaces in which R is partitioned into two blocks Rl and R each of which cover S. Further, each Z line in Rl contains at least three points. Lines then will be of the fi~st type or oeeond type depending on whether they are in Rl or R ' symbols P 1 r,
p
2 r,
In addition, we use the
Z
P l:Z r for any coplanar point and line to indicate that of
the lines through P intersecting r, all are of the first type, none are of the first type, and exactly one is of the second type respectively.
Two distinct
points P and Q, each coplanar with the line r are said to be of the same type with !'espect to
Y',
denoted P
P l:Z rand Q l:Z r.
Q, if P
1 rand Q 1 r,
or P
2 rand
Analogously, the symbols P 1 Q and P
Q are collinear and (P,Q) is in Rl and RZ respectively.
Q 2 r, or
2 Q indicate
that P and
We are now ready to state
our axioms for the partial line space (R(A),F(A)) of an affine space A. DEFINITION: A partial line space (S,R=R ']rassn;ann spac.?
U R ) is termed an abstract affine Z l (AAG-space) if it obeys the following axioms:
(1.1) FOl' ali lines r in R, if P is in S-r then P f r, P---! r,or P - r.
(l.Z) For a l£ne r in R, there exist tl.Jo points P and Q in S-r such that P f Q, P - r, and Q - r. (1.3) For any line r of the
fi~st
type and for any point PES-r skew to r, the
set oj aU po,:nts Q collinear with P and coplanar with r type denoted r(P).
fOml
a Zine of the first
Further, each such coplanarity Q - r is of the
fi~st
type.
627
Characterization of the lines of an affine space
(1.4) For all triples of points P, Q, and T, if P 2 Q and Q 2 T, then P 2 T. (1.5) Let r be a line, and P and Q two points coplanar with r.
Then P and Q
are of the same type with respect to r if and only if they are collinear.
From (1.4) it is easy to see the following. PROPOSITION 1: If r is a line of the first type in an AAG-space and P is coplanar with r, there is at most one point Q E r such that P
P
1~2
r.
2 Q,
1 r or P 2 s.
and hence P
If s is a line of the second type and P - s, then P
1 s or
If A is an affine space of dimension greater than two, it is not difficult to verify the five axioms above showing that (R(A),F(A)) is an AAG-space where R(A) is the set of lines of A and F(A)=R l U R2 where an element of Rl consists of all lines in a plane through a point, and an element of R2 consists of a planar parallel pencil. It is the purpose of this paper to prove the converse: every AAG-space is isomorphic to the Grassmann space of lines of some affine space. We conclude with a result from [1 I whiCh characterizes the Grassmann space of lines in terms of its maximal subspaces.
This characterization will then be
used to prove the theorem stated above by constructing the maximal subspaces from the axiomatized family representing the planar pencils F(A) given above and showing that these maximal subspaces obey the axioms of theorem 2. THEORHl 2: A pa:rtial line space is isomorphic to the Grassmann space of lines in an affine space if and only if its maximal subspaces satisfy the following set of properties:
Al - Three points pairwise collinea:r are in a subspace. A2 - No line is a maximal subspace.
Further, the maximal subspaces are
partitioned into three families L , L2 and P. l
Letting L
(i) (ii) (iii)
= Ll
U L2
we have:
TEL,T'El:l,T~T'
1;
, T ~ T' ~ TnT' = 0; 2 TEE, TI E P ~ Tn TI = 0 or Tn T,T' E
1:
(iv) V r E R, 3! TEL, 3!
TI
TI
E R;
E P such that reT and r c
TI.
628
F. Mazzocca and D. Olanda
A3 - Every point PES belongs to a member of [2' The authors would like to thank Prof. T. Brylawski for his collaboration in translating the present paper.
2. FIRST PROPERTIES OF (S,R) In the following (S,R) represent an AAG-space.
In this section we will
determine the family V of maximal subs paces of (S,R) and give some of its properties. PROPOSITION 1: Given a point P and a line r coplanar with PJ the subset V(P,r)
{QES: Q-PandQ-XforaUXEr}
is a maximal subspace containing P and r.
PROOF: Since every subspace containing P and r is obviously contained in V(P,r), it will suffice to show that V(P,r) is a subspace. tinct points of V(P,r). by (1.1), (A,B)
~
Let A and B denote two dis-
If either A or B belongs to r, then obviously A - B, and
V(P,r). The same result holds if either A or B is the point P.
Thus, we may assume that both A and B are distinct from P and that neither is in r.
Since A and B are both collinear with P, by (1.5) they are each of the same
type as P with respect to r and thus we have A ~ r (1.5).
~
B, and so A - B, again using
If now C is a point on the line (A,B), by (1.1) we have that C E V(P,r)
and this concludes the proof. PROPOSITION 2: Every maximal subspace of (S,R) is of the type V(P,r) for some P and r.
PROOF: Since R is a covering, no point is a maximal subspace. and proposition 1, no line is a maximal subspace.
Further, by (1.2)
Thus, if Wis a maximal sub-
space, it contains at least one line r and a point P ~ r, and thus, by proposition 1, equals V(P,r). From the two propositions above, it is easy to verify the following.
629
Characterization of the lines of an affine space
PROPOSITION 3: Let V(P,r} be a maximal subspace of (S,R). s
~
For every line
V(P,r} and every point Q E V(P,r)-s, one has that V(P,r) = V(Q,s).
It follows
that two distinct maximal subspaces have at most one line in common.
PROPOSITION 4: Every line r is contained in exactly two distinct maximaZ subspaces. PROOF: Let P and Q be two non collinear points each coplanar with r (such a choice is possible by (1.2)). Then V(P,r) F V(Q,r}. containing r and let T be a point of W-r. of the same type with respect to r.
Now let Wbe a maximal subspace
Since P f Q, by (1.5) P and Q are not
From (1.5) and proposition 1.1 it also fol-
lows that P - T or Q - T, and thus T E V(P,r) or T E V(Q,r).
From proposition 3,
one has that W= V(P,r} or W= V(Q,r}. From propositions 1, 3 and 4, the following is easy shown. PROPOSITION 5: GiVen a line r, let V and Wbe the two maximal subspaces which oontain it. (2.1)
Then we have: If P E V-r and Q E W-r, then P f Q.
Denote by the term plane every maximal subspace which contains two lines, one of type 1 and one of type 2.
A maximal subspace which is not a plane is
termed a star and is called proper or improper according to whether its lines are all of the first type or all of the second type. We denote by P, Ll , and L2 respectively the families of planes, proper stars, and improper stars with L = Zl U Z2 the family of stars. We conclude this section with the following proposition whose verification is left to the reader. PROPOSITION 6: If r is a Zine of the first type and P is a point whioh is skew to r, of the two maximal subspaoes passing through the line r(P}, see (1.3), one oontains r and the other P.
630
F. Mazzocca and D. Olanda
3. A CHARACTERIZATION OF THE GRASSrWm SPACE OF LINES OF AN AFFINE SPACE We begin by proving the following. PROPOSITION 1: Ii' r is a line of the second type and V and Wthe two maximal subspaces whi(!h contain it, then one of them is a plane and the other is an improper
PROOF: Choose two points P E V-r and Q E W-r.
One has P t Q by (2.1) and we may
assume that P 1 rand Q '2 r, whence V = V(P,r) E P. that T - rand T - Q, and thus T 2 r by (1.5).
For every T E
~I-r,
one has
We now show that in Wthere are
no lines s E Rl with s n r f 0. Let t be a line of Wwhich does not intersect r.
Choose two points A and B
Then H '2 A, H '2 B, and so A '2 B.
on t and a point H on r.
Thus t E R2 and Wis
an improper star. From the proof above, we have the following. PROPOSITION 2: A ma.r:'£mal subspace containing two distinct lines r and s of the s8eond type with r n s
~
0
ie an improper star.
Since R2 covers St every point belongs to at least one line of R2, and thus from proposition 3.1 we have: PROPOSITION 3: Ever!! point PES belongs to at least two distinct lines of the second type.
PROPOSITION 4: For every point P, the set (3.1 )
a
p
r
UR2
r
E
PEr 7-S
a)1 improper sta!'.
PROOF: Let a and b be two lines of the second type passing through P (cf. proposition 3) and let A and B denote a point of a- {P} and b- {P} respectively.
By
(1.4) one has that A - Bt and letting r denote the line (AtB), we have P - r by (1.1). Since V(Ptr) contains the lines a and b, by prop. 2, V(P,r) is an improper
Characterization of the lines of an affine space
star, whence it is sufficient to prove that 0p If L E r then, obviously L E V(P,r). ~
V(P,r).
= V(P,r). Assume that L is in
0p'
If L ~ r, since L 2 P one has by (1.4), that
L - A and L - B so that L - r by (1.1). 0p
631
It follows that L E V(P,r) and hence
On the other hand, for every point T E V(P,r)- {P}, since V(P,r) is
an improper star, we have that (T,P) is in R2• Thus T E 0p' and V(P,r)
= 0p'
From proposition 4 we have immediately the following: PROPOSITION 5: Two
imp~ope~
stars have empty
inte~section.
PROPOSITION 6: Of the two maximal subspaces which contain a line r E Rl one is a plane and the
is a
othe~
p~ope~
star.
PROOF: If V and Ware the two maximal subspaces through r, let P be a point in V-r Then P f Q by (2.1) and we may assume that P 1 rand Q 1-2r , so that W= V(Q,r) E P. For every T E V-r, one has: T - rand T - P so that by
and Q a point in W-r. (1.5), T 1 r.
This shows that there are no lines of type 2 in V meeting r.
If,
on the other hand there exists in Valine t of type 2 which does not intersect r, denote by Vt the improper star passing through t (see proposition 1). Then V F V and if L E Vt - t, one has L f r (by (2.1)) and r(L)=t. This contradicts t (1.3) since t E R2• Thus V is a proper star and the proposition follows. PROPOSITION 7: Let
TI
be a plane, P a point in
TI,
The following then hold:
TI.
(3.2) If rE R2 and s E R , then rand s (3.3) If
(3.4)
rE Rl and s
The~e
E
2 R2, then r and
a~e
E
disjoint.
S inte~sect.
is at least one line of the second type in
PROOF: Assertion (3.2) follows from corollary 2. s
and r, s two lines contained in
R2 be two lines of
TI.
To prove (3.3), let r
Assume rand s are disjoint.
star passing through s and choose a point T in V-so have T f rand r(T)
TI th~ough
P. E
Rl and
Let V be the improper
Then, by (2.1), one would
= s E R2 which contradicts (1.3).
Finally, we prove (3.4). which, by (3.3) intersect.
Any plane
TI
contains two lines r
E
Rl and s
E
R2
Denote by L the intersection r n s and let Y be a
point of s- {L}. Since, evidently, Y 1~2 r, one has, using (1.5), that X 1~2 r
632
F. Mazzocca and D. Olanda
for every point X E n-r.
If X and H are two distinct points of r neither of which
is L, we have L 1,2 (Y,H) so that, by (1.5), X 1,2 (Y,H).
Assertion (3.4) now
follows. Using proposition 7, the following two propositions are easily verified. PROPOSITION 8: Let r be a line of the first type in a plane n.
For every point P
in n-r, we have P 1:2 r.
PROPOSITION 9: FOi' any line r
Rl and point P E S-r skew to r, the line r(P) is
E
contained in the pl'oper star passing through r.
PROPOSITION 10: FO!' every subspace a in Ll and every point P r;. a, there is a line t coplanar with P and aontained in o.
PROOF: Let s be a line of o. P f s.
If P - s we are done so assume that either P ~ s or
If P f s, the line s(P) is contained in a by proposition 9 and is coplanar
with P.
If, on
the other hand, P ~ s, let A be the unique point of s collinear
with P and let r be a line of a not passing through A.
Since P r;. a we have
P ~ r or P f r and in either case the assertion follows. PROPOSITION 11: GiVen a line r pass"ing through r.
E
Rp let V and Wdenote the two maximal subspaces
Any maximal subspaae L which meets r in a unique pain t P in-
terseats either V or W in a line through P.
PROOF: Let T be a point of L- {P} and suppose, for example, that V E Ll and (see"proposition 6).
If T .2.'1:"1"e
pal'aZZel then the !'emaining q2 lines f01'lT/ a basis of the 7l. - !'Ol,) 2
PROOF: First we prove that each vector of length q and weight 1 is an element of the rowspace of A.
Indeed if a vector of weight 1 has its unique non zero compo-
nent on the column associated to the point Pi then we have to prove that there exists a set Pi
L
of rows (or lines) such that its sum is this vector or such that
is the unique affine point which is incident with an odd number of elements of If we denote the set of all affine 1ines incident with a point p (which is an
affine point or a point at infinity) with [pI, and if m is an arbitrary point at infinity then L.=(Pi]+[m], i.e. the set of all affine lines which are incident with Hence the 7l. 2-rank of A is q2. If {m 1 ,m , ... ,m q+ } is the set of the points at infinity of the affine plane, and 1 2 if {L ,L , ... L ,L ) is a set of q arbitrary affine lines such that m,.IL,. 1 2 q-1 q (i=1,2, ... ,q) then il=(set of all affine lines} - {L ,L , ... ,L q } is a basis of the 1 2 7l.2-rowspace of A. Pi or with m but not with both has this property.
Indeed L.=(fm.]-L.)+[m , 1 , q+ 1] (i=1,2, ... ,q) and since ([m,.]-L,·)9 and [m lIe,:, and since the q2+q affine lines generate a ?l.2-linear space of dimension q+ q2, we have that to is a basis of the 7l.2 -row space of A. Hence also each of the 2(q2) sets of affine points is the sum of a set of lines of to in a unique way.
14.
THEOREM: Let L lie a line in PG(3,q) with q odd.
e"ntain~ng
If
l:
is an a!'bit!'a!'y line system
no lines inte!'secting L then the!'e exist exact ly 2 (q2) line systems
LI
:1c'1ls-isting only of lines intersecting L such that HL'EZ1(n,q,7l.2)11Z1(n,q,7l.2)' v';;~:ne
1
r:ds p!'ope!'ty one can construct a basis fo!' Z (n,q,7l. ) and Zl(n,q,7l. ).
2
PROOF: Consider in the quadratic extension PG(2,q 2 ) of PG(2,q) a plane that ·"onPG(3,q)=L.
2
TI
o
such
Then there exists a 1-1 correspondence between the q4 points
647
Zz,cohomology of projec five spaces of odd order
-{L} and the q4 lines of PG(3,q) not intersecting o (where L denotes the quadratic extension of the line L). Consider now an arbi-
of the affine plane AG(2,q2)=rr
L
trary pOint mEL-L and the q2 lines of AG(2,q2)=rr -{L} with m as a point at infini-
o
ty: {D .0 •.•. Dq 2}. Then the Baersublines of such a line 0iu{mJ form a Miquelian 1 2 inversive plane of order q and so the Baersublines containing the point m form on the q2 affine points of O. a Oesarguesian affine plane of order q. 1
Since the Z2-rank of an affine plane of order q is q2 there exists a basis 2
of q2ZZ2-linear independent Baersublines {B~B~ ... B~ } containing m. such that each 11 1 point subset AcO. is the sum of these q2 basis Baersublines (i=1.2 •...• q2). - 1 Since the affine lines {O .0 •... ,0 } form a partitioning of the q4 affine _ 14 2 q poi nts of rr - {L}. each of the 2G sets of affi ne poi nts is tne sum of such bas i c o _ q2 q2 . Baersublines. Or if (lCn -{L} then a=.1: .1: ,\. ,B~ with A. .ElI. • Now with each - 0 1=1 J'" 1 1J 1 1J 2 (lCrr -{L} corresponds uniquely a set of lines of PG(3.q) containing no line inter- 0 secting L and conversely.
L-L
With a Baersubline containing a point m of
sponds uniquely a regulus in PG(3.q) containing the line L (see [41).
corre-
Hence the
transversal of this regulus form a regulus whose q+l lines all intersect L.
We
denote the regulus, the set of transversals and the hyperbolic quadric which is their union associated to the Baersubline B~ respectively by I:~. (I')~ and ..
.
J
1
H\ I:~+( I:')~ (i .j= 1,2 ..... q 2). 1
zl(3 , q2'71 • 2 )()Z (3 .q2'71 , 2) •
1
1
.
1
Of course the hyperbolic quadric H~ belongs to
Now if
J
I:
is a line system of PG(J,q) containing no
q2 q2 j lines intersecting L and if I()rr =(l = , l: .E A. ,B ., then we have that o 1=lJ=1 1J 1 q2 q2 j 1:= . I: . I A,· 1: .. 1=1 J=l 1J 1 q2 q2 j Then, if we denote i~l j~l A ( I')i by I', U q
2
E+ E' = . E
q
2
i 1 . E A,. H.El. (3, q ; 7l ) () Z (3. q ; 7l ) • 2 2 1
1= 1 J= 1 1J J
Hence for each line system containing no lines intersecting L we have constructed a line system E' whose line all intersect L such that Zl(3,q;7l )()Zl (3,q;7l ). If now I" is another line system whose 2 2 lines all intersect L such that also 1: + E" E Zl(3,q;7l 2)()Zl(3,q;7l 2) then
I:
+
E' E
Zl(3,q;7l ) () Zl(3,q;71 ) and so E" = E' + f; where 1;, is a line 2 2 system whose lines all intersect L and such that I; belongs to E' + E" E
Z'(3,q;7l )()Z(3.q;7l ). We denote N=U;1I1; is a line system whose lines all inter2 2 1 sect the line L and such that i;=Zl (3,q;7l ) nz (3,q;7l 2). We denote the q+1points 2
w. Mielants and H. Leemans
648
on L by P ,p , ... ,p and the q+l planes incident with L by 'IT ,'IT , ... ,'IT Also 1 2 q+ 1 1 2 q+ 1 we denote by [p.y..] the set {MUM is a line of PG(3,q),p.IMIn.} and by T . . the 1 1
1
J
1,J
line system[Pq+1,1Tq+1]+[Pi,'ITq+1J+ [Pq+l,'IT j ]+[Pi''IT j ] (i,j=1,2, ... ,q). If sEN then ~n1T. is a set of affine lines of the affine plane 'IT. {l}, such that each affine J J point is incident with an even number of lines of r;im. (since sEZ (3,q;~2». ~ J 1 Since the linear map a:~2 +q~~~ associated to the incidence matrix of points and lines of an affine plane of order q has ~2- rank q2 we have that dim ker)= q. q The 2 line systems of the affine plane n.-{L} with this property are then of the q±l J q+ 1 form .~1 m.. [p.,-rr.] such that m.. E~2 and .E m.. =O (mod2). If we denote the 1=
1J
1
J
1J
1=1
1J
set of all lines of PG(3,q) incident with a paint p by [p] then we have that if Pi is an arbitrary point incident with L that sf'( Pi] is a set of lines of [Pi] such that each plane incident with p. and not incident with L is incident with an 1 q even number of lines of t;n[Pi] (since sEZ1(3,q'~2». So (dually) the 2 line systemsof[Pi] with this property are of the form
jt~
mi}Pi'IT j ] with
it~
mi{O
(mod2) . Hence the
~2- linear space of line systems of Zl(3,q;7lz)nZ1(3,q;~t such
that each line intersects the line L. is the matrices (m .. ) over
linear space of (q+l)x(q+l)
~2
with the property that each row and each column has even 2 q q q The number of such matrices is 2 .2 .... 2 (q factors) or 2(q ). 1J
weight.
~2-
A basis of this ~2- linear space of dimension q2 are the q2 matrices (E .. ) (i,j=1,2, ... ,q) where 1J 1 if (u,v)E{(q+l,q+l),(q+l,j).(i,q+l)(i,j)}
o othenlise With each matri x E., corresponds ali ne system T. .. .
1.J
1
{H~lIi,j=1.2, ... ,q2}U{T J
Hence the set 1
u,v=1,2 •... q} is a basis of Z (3,q;~2}n Zl (3,q;~2)·
u,v By adjOining a line system [p] = {lEQ1 (3,q)lIpIl} for an arbitrary point p
or ali ne system [ 'If] = {lEQ1 (3 ,q) 1111 'If} for an arbi trary plane tivelya basis of ~2-
Zl(3,q;~2) and of Zl(3,q;~2).
Hence if
'If,
then we have respec-
zlEZ1(3,q;~2) is a
cycle or a line system of PG(3,q) such that each point is incident with an
even number of lines of zl then q2
q2
z:=.[ .L A •• : 1=1 J=l 1J
.
H~+ 1
q
q
E l: IJ V U=l U=l u
T
U.V
+\I[n]
and
Z 2oCohom%gy of projec tive spaces of odd order
A. ., \l ',J
649
, 1I=.71. 2 • u,v l 1 Dually if Z EZ (3,q;Z2) is a 7l. - cocycle or a line system of PG(3,q) such tha 2
.
1
that each plane is incident with an even number of lines of Z then q2 q2 i q ~ zl=.l: l: \l T +v[p] with A'·J·,\luv'lI=. 7l. 2 • , =1 J.l:=1 A,..J H.+ J U= 1 V= 1 Uv U v I..
15.
PROPOSITION 9: The Zines of a hyperboZia quadria form a non triviaZ (whiah is not a
~2-
~2-
aoayaZe
aoboundaryJ.
PROOF: Since each plane intersects a hyperbolic quadric in 0 or 2 lines, the hyperbolic quadrics are 7l. 2- cocycles. (Dually since each point is incident with 0 or 2 lines of a hyperbolic quadric they are also 7l. 2- cycles). Since the set of 7l. - coboundaries of PG(3,q) is of course invariant under the group of all collin2
eations of PG(3,q), and since prL(3,q) is transitive on the hyperbolic quadrics either no hyperbolic quadric is a
~-coboundary, or all of them are. Now we have 4 proved that there exi sts a set of q independent hyperbo 1i c quadri cs (whi ch form a
subset of a basis of Zl(3,q;71.2)nz1(3,q;71.2); and so if we denote the linear subspace of Zl(3,q;71. 2) spanned by the hyperbolic quadrics by W, then we have dimw;..q4. But dimB 1(3,q;71. ) =q3+q2+q. Hence ~1(3,q;71.2) and so wnB1(3,q;71.2)=O. 2
16. SOME OPEN PROBLEMS
1.
The determination of the 7l. 2- cohomological dimension of the Beutelspacher para 11 e1ism.
2.
The classification of all regular l-graphs of PG(3,q) (q odd) which are not
trivial (or which are not 7l. - coboundaries). 2
3.
The natural representation of PGL(n,q) on the rational top homology space of
the corresponding building K (of type An) : Hn_1(K,Q) is the interesting Steinberg representation.
What can be said of the natural representation of PGL(n,q) on the t 7l. - cohomology spaces H (n,q,71. ) if q is odd? 2 2 4. If G.q:>rL(n,q) then the G-equivariant 7l. - cohomology space of degree 2 t:H t (n,q;71. 2;G) is the 7l. 2- cohomology space of degree t of the 7l. 2- cochain sub-
650
W. Mielall/s alld H. Leemalls
complex of G-invariant t-systems of PG(n,q) ators are defined by taking restrictions.
(O~t~)
where the
~2-
coboundary oper-
If q=l and if G is a (t-1)-transitive
group of permutations of degree n+1 then it has been proved (see [6J) that if ~
t=2:H~(n, 1 '~2;G)
is the
~2-
vectorspace whose vectors are the G-invariant bipar-
titionings of this (n+l)-set and if
~3
we have proved (see [6J that
Ht(n,1'~2; G)#O if and only if the permutation group G can be obtained as a transitive extension from its stabilizer group by the Generalized Graph Extension 1heorem of E.
Shult and D.G. Higman.
For t=3 many doubly transitive permutation groups of Cheva11ey type can be obtained in this way but also the doubly transitive representations of degree 176 and 276 of the sporadic simple groups of Higman-Simms (H.S.) and of Conway (.3) have the property that thei r equi vari ant trivial.
~
-cohomo logy spaces of degree t=3 are not
For t=4 this is also the case for the affine group AGL(3,2) and for the
Mathieu group M1l in its exceptional 3-transitive representation of degree 12 and for t=6 for the Mathieu groups Ml2 and M24 in their classical 5-transitive represel1tations. An interesting open problem is: what can be said of the geometrical meaning t of the equivariant ~-cohomology spaces H (n,q;Z2;G) of a group G~PrL(n,q) in the case that q>l?
BI BLl OGRAPHY 1. 2. 3. 4. 5. 6. 7. B.
P.J. Cameron, Generalisation of Fishpr's inequality to fields with more than one element, Combinatopies (Proc. Brit. Comb. Conf. 1973), London Math. Soc. Note Series 13 1974. P.J. Cameron, Automorphisms and Cohomology of Switching Classes, .c :·~:Y'.~t equality holds iff 0 is an (r,A)-design,
Inequalities for point stable designs
659
the second equality holds iff D is a PSI(a,r).
PROOF: From t2 =
y-- (w(p)
+ [ pJ 2) and Thm. 2.3. we obtain the first inequality. peP The second inequality is equivalent to the variance-inequality (compare Thm. 2.1.)
y-- [pJ 2 >
0
2 v- l with equality iff [pI
= 0 v- 1 for all points p.
peP
5. BOUNDS FOR THE CONNECTION NUMBERS OF REGULAR POINT STABLE DESIGNS The following theorem generalizes and improves the inequalities of AGRAWAL [1,2J for the intersection numbers [B,CJ of different blocks B,C of a l-design. The AGRAWAL-inequalities contain the SHAH-inequalities for PBIBD's [ 11) , and the CONNOR-MAJUMDAR-inequalities for 2-designs [5,7] . THEOREM 5.1: Let D be a connected PSI(a,r),
Then for any two different points p,q we have -1
max {2r - b, r - P , 2(a - p)v m l min {r - Pm' 2(a
Pl)v
-1
+
+ P - r} m
~
[p,q]
~
Pl - r}.
The proof will be given elsewhere. COROLLARY 5.2: (AGRAWAL [1,2]): Let D be a connected l-(v,k,r)-design, {rk, Pl'"'' Pm} , rk > Pl >"'>P m' Then for any two different blocks B,C we have specN
=
-1 max {2k - v, k-P 1,2rkb
min {k, 2(rk
Pl)b
-1
-k}~[B.C]~
+ P - k}.
1
REMARK: It is easily seen that the application of Thm. 5.1. to the dual of the 1design D in Cor. 5.2. yields exactly the AGRAWAL-bounds if det(ATA) = 0, where A is the incidence matrix of 0, while otherwise both bounds 2rkb -1 - k ~ [S,C] .;;; k are improved by Thm. 5.1.
660
A. Neumaier and K.E. Wolff
BIBLIOGRAPHY 1. 2. 3. 4.
5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
H.L. Agrawal, On the bounds of the number of common treatments between blocks of certain two associate designs, Cal. Statist. Assoc. Bull., 13 (1964), 76-79. H.L. Agrawal, Comparison of the bounds of the number of common treatments between blocks of certain partially balanced incomplete block designs, Ann. Math. Stat., 37 (1966), 739-740. R.C. Bose, W.G. Bridges and M.S. Shrikhande, A characterization of partial geometric designs, Discrete Math., 16 (1976), 1-7. R.C. Bose, S.S. Shrikhande, N.M. Singhi, Edge regular multigraphs and partial geometric designs with an application to the embedding of quasiresidual designs, Colloquio Internazionale suZIe Teorie Combinatoroie, Torno I, 49-81, Accademia Nazionale dei Lincei, Roma 1976. W.S. Connor, Jr., On the structure of balanced incomplete block designs, AmI. Math. Srat., 23 (1952), 57-71. H. Hanan;, On transversal designs, in "Comb ina toroics , Parot l", 42-52, Math. Centre Tracts No.55, Math. Centrum, Amsterdam, 1974. K.N. Majumdar, On some theorems in combinatorics relating to incomplete block designs, Ann. Math. Stat., 24 (1953), 377-389. D. McCarthy and S.A. Vanstone, On the structure of regular pairwise balanced designs, Discrete Math., 25 (1979), 237-244. R.C. Mullin and S.A. Vanstone, On regular pairwise balanced designs of order 6 and index 1, Uti L Math., 8 (1975), 349-369. A. Neumaier, t 1 - designs, J. Comb. Th., A28 (1980), 226-248. S.M. Shah, Boun~s for the number of common treatments between any two blocks of certain PBIB designs, Ann. Math. Stat., 36 (1965), 337-342. K.E. Wolff, Punkt-stabile und semi-partia1-geometrische Inzidenzstrukturen, Mitt. !·!ath. Sem. Giessen, 135 (1978), 1-96. K.E. Wolff, Rank classification of point stable designs, (to appear). K.E. Wolff, Point stable designs, Finite Geometries and Designs, London Math. Soc. Lecture Notes 49 (1981), 365-369, Cambridge Univ. Press. K.E. Wolff, Strong point stable designs, Geometries and Groups, Proceedings Ber1 in 1981.
Institut fUr Angewandte Mathematik der Universitat Freiburg Hermann-Herder-Str. 10 D 7800 Freiburg Federal Republic of Germany
Hein-Heckroth-Str. 27 D 6300 Giessen Federal Republic of Germany
Annals of Discrete Mathematics 18 (1983) 661-666 © North-Holland Publishing Company
661
ON THE STRUCTURE OF TRANSLATION GENERALIZED QUADRANGLES Stanley E. Payne
ABSTRACT A technical restriction is given on the structure of a translation generalized quadrangle of order (s,t).
Two corollaries are: if t is even then
s = t or t = s2; the translation group is the full set of elations. 1. INTRODUCTION The definition of translation generalized quadrangle (TGQ) given by Thas [6] for a generalized quadrangle (GQ) of order (s,t) with s = t was generalized in [2] to the case s
~
t.
Considerable progress was made in [4] on the study of TGQ, and
this note may be considered a sequel to that article.
We collect a few results
from [1], [4], [7] and use them to derive additional information about the structure of TGQ.
The notation and terminology are that of [4] and [7], and for the
most part will not be repeated here. It seems possible that a TGQ of order (s,t) must have either s ~ t or t = s2. We have not been able to prove this for odd t, but our techniques do yield a new proof that if (S(p) ,G) is a TGQ of order (s,t), s F t, then for some prime power q and some odd integer a, s = q~t= qa+l.
The rather involved, partially geometric
proof in [4] is replaced by an essentially equivalent one using only combinatorial algebra.
Our main technical lemma is that any triad of points containing the base
point p must have either 0 or 1 + tis centers.
It follows that the translation
group G is the full set of elations about p and that the multiplicative group of the kernel is always isomorphic to the group of homologies about p with given center y, y f p. 2. COLLECTED RESULTS Let S = (P,L,I) be a GQ of order (s,t), 1 < s, 1 < t.
For a fixed pair
of noncollinear points let n.1 be the number of triads (x,y,z) having exactly i
(~y)
662
S.E. Payne
centers, 0
~
~
i
1 + t.
The following identities are equations (1), (2), (3) of
[ 7).
l+t (1 )
E
i=O
2
s t - st - s + t
n.
1
l+t (2)
2
z in.1 ;=0
t s - s
l+t (3)
1:
i (i -1) n.
1
i=O
Divide (2) by
5
to get (2') and (3) by t to get (3').
Then subtract (2') from (3')
to obtain the following: ( 4)
l+t
2
z (si - i(s+t)) st
i=2
nl ·
ni
The coefficient on n in (4) is nonnegative if and only if i equals 0 if and only if i = 1 + tis. This yields the following: (5)
No
t~iad
~
1 + tis, and
containing (x,y) has more than 1 + tis centers if and only if each
triad has exactly 0 or 1 + tis oenters if and onZy if no such triad has i oenters with 0 < i < 1 + tis.
in this oase nO
= t(s-1)(s2_ t )/(s+t).
This result is a nice completion of the result of [7) that for s if and only if either t
~
t n l
=
= 1 or s = t and (x,y) is antiregu1ar.
Let e be a collineation of S.
-
Let f (respectively, f) denote the number of
points (respectively, lines) fixed by s, and g (respectively, g) the number of points x for which x6 ~ x but x6 - x (respectively, lines L for which L6 ~ L but La - L). (6)
The following is contained in 111.1 of [3).
(l+t)f + 9
0
(l+s)f+
g.
Benson [ 1) proved a related useful result.
On the structure of translation generalized quadrangles
663
(1+t)f + gEl 1 + st (modulo s+t).
(7)
(The preceding result can also be obtained from computations in [3) . However, [3) contains an error pointed out by Mr. Klemins Skorka [5).
The reason-
ing leading to eq. (36) of [3) claims to depend only on the hypothesis that e have prime order n, but the argument is faulty in case n = 2 and s = t, and in any case was originally suggested by Benson's work.) If e is a nonidentity symmetry about some line, then f = §
= O. (6) and (7) apply, forcing st(l+s)
are powers of the same prime, s
~
~
0 (modulo (s+t)).
t, a result of Benson a
[1)
a±l
prime power q and a positive integer a, s = q and t = q
given forces 1 + q to divide 1 + qa, so that a must be odd.
1 + t + st and If also sand t
says that for some The congruence just This proves the
following: (8) If S has a nonidentity symmetry about some line, then st(l+s)
(s+t)).
~
0 modulo
If in addition s and t are powers of the same prime, there is a prime
power q and an odd integer a for whiah s
= qa
and t
= qa+l or t = qa-l (if s ~ tJ.
Of course there is a dual result for symmetries about points. and t are powers of the same prime, s
~
Hence if s
t, then S aannot have both nonidentity
symmetries about points and nonidentity symmetries about lines.
3. APPLICATIONS TO TGQ Suppose that (S(p) ,G) is a TGQ. tions of
This means that G is a group of col linea-
fixing each line through the point p and acting regularly on the set 2 p - pI of s t points not collinear with p, and G contains a full group of s S
symmetries about each line through p, so s
~
t.
It is shown in [4) that G is
elementary abelian so that sand t are powers of the same prime.
Hence the fol-
lowing result of [4) is an immediate consequence of (8). S has parameters (s,t)
(9)
if s
~
= (qa, qa+l), where q is a prime power
and a is odd,
t. It is shown in [4) that if e is any element of G which is not a symmetry
about any line through p, then for some integer r, 0
~
r
~
1 + t, there are r
664
S.E. Payne
lines through p such that the fixed points of e are p together with all the other points on those r lines, i.e. f = 1 + rs. through P. so f = 1 + t. easy to see that g parameter g or (10)
6
are just the lines
If r = 0 it is easy to see that 9 = O.
= (t+l-r)s. From
(6)
If r f. 0, it is
it is easy to solve for the remaining
g. Then an application of (7) yields
r '" 0 (modulo l+q) Let x be a fixed point of P - pl. 6
unique e E G with x = z. G
The fixed lines of
For each point z
E
P - pl there is a
From (4) we know that (p,x,z) is a triad if and only if
is not a symmetry about some line through p, in which case the number of centers
of (p,x,z) is the number r of lines of fixed points of
6.
Hence we can apply
(l), (2), (3) to the fixed pair (p,x). and (10) implies that ni = 0 unless i '" 0 (mod l+q). In particular, n. = 0 for 0 < i < 1 + tis, so from (5) we see that 1
precisely t{s-l)(s2_ t )/{s+t) tl,iads containing (p,x) aroe acentr'ic, and the remaining (t2-1)s2/{s+t) such triads have exactly 1 + q
= 1 + tis
Since p
center's.
belongs to no unicentric triad, by 2.l.{iv) and 6.3 of [4). G is the
fu~l
set of
Moreover. by 6.4 the multiplicative group of the kernel of
elations about p.
(S{p) ,G) is isomorphic to the gr'Oup of al~ collineations of S fixing all lines 1
thr'ough p and fixing some point y, yEP - P .
If t is even, the number of centers of any triad containing p must be odd by 2.2 (i) of [4), which implies the following, since no = O. (11) If s{p) ,G)
is a TGQ of order' (s,t), 1 < s < t with t even, then t
We mention one last corollary.
= 52 .
Let S be a GQ of or'der (s,t), 1 < s, 1 < t.
Suppose S satisfies (M)
for a point p such that each tr'iad containing p has at
least
S{p) is a TGQ and t = s2.
p~o
centers.
The~
Let the reader supply a proof starting with 7.7.(ii) of [4).
BIBLIOGRAPHY 1.
C.T. Benson, On the structure of generalized quadrangles, Jour'. Alg., 15 (1970), 443-454.
On the structure of translation generalized quadrangles
2. 3. 4. 5. 6. 7. 8.
665
S.E. Payne, Generalized quadrangles as group coset geometries, Congressus Numerantium 29, Proc. 11th S.E. Conf. Comb., Graph Theory, Comp., (1980), 717-734. S.E. Payne and J.A. Thas, Generalized quadrangles with symmetry, Simon Stevin, 49 (1975-1976), 3-22 and 81-103. S.E. Payne and J.A. Thas, Moufang conditions for finite generalized quadrangles, in P.J. Cameron et al. (ed.) Finite Geometries and Designs, London Math. Soc. Lecture Notes 49 (1981), 275-303. K. Skorka, 1979, private communication. J.A. Thas, Translation 4-gona1 configurations, Rend. Aaaad. Naz. Linaei, Serie VIII, vol. LVI, fasc. 3 (1974), 303-314. J.A. Thas and S.E. Payne, Classical finite generalized quadrangles: a combinatorial study, Ars Combinatoria, 2 (1976), 57-110. J.A. Thas and S.E. Payne, Generalized quadrangles and the Higman-Sims technique, European Jour. Comb., 2 (1981), 78-89.
Department of Mathematics and Statistics Miami University Oxford, OH 45056 USA
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Annals of Discrete Mathematics 18 (1983) 667'{)94
667
© North-Holland Publishing Company
SUR LES K-ARCS COMPLETS DES PLANS DE GALOIS D'ORDRE IMPAIR Giuseppe Pellegrino
INTRODUCTION Soit PG(2,q) un plan de Galois d'ordre q impair, c'est-a-dire un plan projectif construit sur le corps de Galois GF(q) (q impair); un k-aPa (ou bien arc d'ordre k) de PG(2,q) est un ensemble H de points du plan ayant les proprietes suivantes: i)
ii)
(HI = k
trois points que1conques de H sont independents (c'est-a-dire i1s ne sont pas alignes).
Nous dirons qu'une ligne droite du plan est seaante (ou aopde), tangente, extepieure a un k-arc H selon qu'elle ait respectivement deux, un, aucun point en
commun avec H. Un k-arc est dit aompZet s'il n'est pas un sous-ensemble propre d'un (k+l)-arc; ou bien, d'une
fa~on
equivalente, si tout point du plan appartient a
une corde au moins de l'arc. La theorie des k-arcs est un domaine de recherche de la part de nombreux Auteurs car le sujet est interessant soit en
1ui-m~me,
soit en vue d'applications
possibles (cf. p.e. [71, chap. 15). Un des problemes centraux de la theorie est le suivant: "Un entier positif k etant donne, etab1ir s'il existe en PG(2,q) des arcs complets d'ordre k". A ce sujet, le resultats suivants sont fondamentaux (Qvist, Bose, Segre; cf. [21, [ 121 , [151 ): 1) Pour un k-aPa de PG(2,q),q impaip. on a k
~
q + 1.
2) En PG(2,q),q impaip. tout (q+l)-aPa -qui est dit aussi ovaZe- est une aonique ippeduatibZe. et invepsement.
Les resultats concernant 1 'existence et la construction d'arcs complets qui ne sont pas ova 1es sont assez recents (cf. [21, [51, [61, [ 181) et peuvent etre
668
G. Pellegrino
resumes de la
fa~on
suivante:
3) En PG(2,q),q impair', existent des aras aompZets d'or'dl'e (q+9)/2,
(q+7)/2, (q+5)/2, (q+3)/2. Ces resultats, pour obtenir lequels on a utilise les methodes les plus variees, ont un caractere general, en ce sens que, sauf tres peu de cas, ils sont valables pour tout q compatible avec 1 'ordre du plan de Galois. Dans cette note nous signalons 1 'existence en PG(2,q) d'arcs complets d'ordre k
~
(q+l)/2 et nous en donnons la construction. Le procede suivi en ce
cas, meme s'il s'inspire des methodes precedemment appliquees, differe remarquablement de celles-ci parce qu'il reconduit 1 'etude des k-arcs a un probleme arithmetique qui n'est pas du tout resolu (cf.n.5).
Par consequent, les arcs ici
construits se referent seulement a des valeurs particulieres de q; mais a 1 'aide d'un calculateur, le procede peut donner des informations tres utiles dans 1 'etude des k-arcs de PG(2,q), meme pour des valeurs de q pas trop petites.
Les
resultats obtenus sont indiques dans la table suivante ordre q du plan
23 29 31
37 41
ordre k de l'are complet
12 14 16 18 18 22 22
43 47
Nous remarquons enfin que par ce meme procede on peut retrouver aussi des resultats de earactere general deja eonnus.
1. PRESENTATION DU PROBLEME
Nous noterons C une eonique (non degenere) du plan PG(2,q).
Renvoyant a la
bibliographie pour tout ce qui concerne les proprietes des eoniques dans les plans de Galois (ef. en particulier (1), [5), [15), [16]), nous ne rappelons iei que des resultats auxquels nous nous rapporterons directement par la suite. i) On a h q = P = 4t .:. 1
(1. 1)
avec p nombre premier (impair), qui est la caracteristique du corps, et h entier pos iti f. ii) lei
=
q+l.
iii) Soit P un point du plan, non situe sur C.
Si Pest interieur a C, i1 Y
passe (q+l)/2 droites secantes et (q+l)/2 droites exterieures a c; si, au
669
Sur les k-arcs complets de PG(2,q),q impair
contraire. Pest exterieur
a c.
il y passe deux droites tangentes. (q-l)/2
droites secantes et (q-l)/2 exterieures
a c.
iv) Les couples (U.V) de points de C. alignes avec p. sont des couples de points correspondant dans 1 'involution. 0p. sur C ayant pole en P. v) Soit Z une droite non tangente
a c et
soit P un point de Z.
Lorsque P
parcourt {Z-C}. les involutions 0P engendrent un groupe L (le groupe des Z
projectivites axiales. ayant axe Z) qui est diedral. Nous noterons d le sous-groupe commutatif. d'index 2 en LZ; alors 6 Z est Z cyclique et. de plus IdZI = q+l. si Zest exterieure 16Z1 = q-l. si Zest secante
a c.
vi) Si la droite Zest tangente
ac
(1.2)
a c.
les involutions 0P sur C. ayant le pole
sur les pOints de Z. engendrent un groupe. que nous appellerons encore
d
' Z
Les
proprietes de d seront discutees par la suite. Z Nous eonsidererons PG(2.q) 1 'extension du plan affine A(2.q); cela nous conduira
a denoter
selon les
r~gles
K' = {K u
oo}.
oil
le point Pea. ~. y) de PG(2.q). lorsque est y ~ O. par P(a.B)
bien connues. Q)
Nous aurons done
= z/O pour tout 0
~ Z E
a considerer
1 'ensemble
K (cf. [121 • p. 97).
Les elements du sous-corps ZpC GF(q) de K seront denotes par O.l ••••• p-l oil 0 et 1 sont le zero et l'unite de K. 11 est utile de rappeler que vii) Si q=4t+l. elements opposes de K ont le
m~me
earactere quadrati que;
viii) Si q=4t-l. elements opposes de K ont caractere quadrati que different. Nous ecrirons z = [J
•
z = 6 en indiquant qu'un element z de K est.
respeetivement. un carre. un non carre. En introduisant en A(2,q) un systeme de coordonnees approprie. la conique C peut etre representee par 1 'equation
2 2 x+y=l,
( 1.3)
ou bien. en forme parametrique, (1.4 )
670
G. Pellegrino
au point (1.0) de C correspondant la valeur u
= du parametre. ro
Soit P(a,8,y} un point du plan, non situA sur C.
Deux points (de
coordonnees paramAtriques) u,v de C sont alignAs avec P si et seulement s'ils se correspondent dans 1 'involution 0p sur C ayant le pole en P; c'est-a-dire si et seulement s'ils satisfont a l'Aquation Bu - (Y+[-2-J elements arbitraires reproduit 1 'ensemble A par somme. DEMONSTRATION: On observe avant tout que la congruence 2x
n (mod m) ales so-
lutions suivantes selon que m et n sont pai rs ou impairs: (
m impair
t
I x = n/2 = r x = (m+n)/2
si n = 2r si n
2r+l
m pair Apres cela, considerons 1 'ensemble T des couples (a.,a.) d'elements distincts de A 1
tels que a.+a. = n (mod m). 1
J
J
Pour ce que nous avons observe, n etant fixe, on a
i) ITI m (m-2)/2 si met n sont pairs; ii) ITI = m/2 si m est pair et n impair; iii) ITI = {m-l)/2 si m est impair, quelle que soit la parite de n.
684
G. Pellegrino
Dans le cas i), formons Z en prenant k=2(m-2)/2+1
(m+4)/2 elements arbi-
g
traires de A; alors, compte tenu des valeurs de IAI et de ITI, l'on voit facilement qu'il existe au moins un couple (a.,a.) d'elements distincts de Z tels que 1
a.+a.=n. J
1
la
fa~on
suivante: l = {r,r+m/2}
U
m/2 elements ai (i=l,2, .• ,m/2).
A', ou A' est un sous-ensemble de A contenant
11 est ITI= (m-2/2.
(a!,a!) d'elements de A' 1 'on a a! + a! 1
ments
J
J
1
3.
1
J
(En effet, n=2r etant fixe, au cas le plus restictif, formons Z de
F n (mod m), en A existent (m-2)/2 ele-
(a. F r, r+m/2) tels que a.+a! 1
1
IAI = 2 + 01/2 + (m-2)!2 = m+l > 01).
Or, si pour tout couple
n (mod m) ou a!
1
1
E
A'.
14ais alors on a
Tout cela vaut pour chaque n.
Dans les cas ii), iii) il suffit de prendre respectivement k =
III = m/2
+
1 = (m+3)/2
;
k = IZI = 1
+
(m-l)/2 + 1
(m+3)/2.
La proposition est ainsi demontree. Dans le but que nous proposons, nous avons interet bles Z ayant une puissance k < [m~4l.
Des solutions
a ce
a determiner
des ensem-
sujet sont indiquees
dans les paragraphes suivants. 5.2.
Si m est une puissance d'un nombre premier, A est un corps.
Formons
les sous-ensembles suivants de A: Z
{v ,Xi
Z
{x·1 I OF x.1 = O}
I V=!l,OFX.= 1
O}
si m=4t-l
(t > 2
IZI=(m+l)/2)
si m=4t+1
(t > 2
IZ!=(m-l)/2)
alors les ensembles l reproduisent A par somme. DEMONSTRATION: Elle est une consequence immediate des proprietes vii), viii) du n.l et dufait que dans un corps fini tout element est 1a somme de deux carras (ef. 14 I, p.l).
5.3.
Soit maintenant m=d'd", ou d"F2.
Si nous considerons A comme un
anneau, A(+) est un groupe abelien et 1 'ensemble G={O,d' ,2d', •• ,(d"-l)d'} est un sous-groupe d'index d' dans A.
Soient
G.1 = G + z.1 (i=O,l, .. ,d'-l ; zO=O) les classes laterales de A par rapport
a G et
(5.1 )
formons 1 'ensemble Zt de puissance
685
Sur les k-arcs complets de PG(2,qJ,q impair
d'+d"-l, y mettant les elements de G et un seul element de chaque classe laterale. L'ensemble Z reproduit A par somme. DEMONSTRATION: Elle est une facile consequence des proprietes des groupes. 5.4. Soit enfin m=2d.
L'ensemble G={O,d} est encore un sous-groupe de A,
d'index d, et la repartition de A en classes laterales par rapport
a G est
la
suivante G. = {i, d+i } 1
(5.2)
(i=O,l, •• ,d-l).
Ainsi sont definis les ensembles Dl ={l,2, •• ,d-l}, D2={d+l,d+2, •• ,2d-l} pour lesquels on a la propriete suivante: 1 'oppose, modulo m, de chaque element de 0
1
appartient
a O2 et
inversement.
Cela
dit, considerons le sous-ensemble Z constitue par les elements 0, d et par un (seul) element de chaque classe laterale (5.2). type z,m-z (et appartenant somme.
a des
Si en Z on a deux elements du
classes laterales distinctes), Z reproduit A par
Dans le cas contraire, Z reproduit tous les elements de A sauf 1 'element
O. 5.5. On peut obtenir une autre solution du probeme 5.0, dans le cas . 1·ler m=p 2h -, 1 en me tt an t en Z au maXlmum . k 3ph- 1 e-1 emen t s de A• En effet partlcu (cf. [llJ, n.28, p.308) de tout plan affine (en particulier arguesien) on peut deduire un ensemble de ph entiers al,a2, •• ,aph tels que, si (ph+l) d = a.-a. a une et une seule solution. 1
J
%d,
l'equation
h
Donc 1 'ensemble {+a '+2, .• ,+a h} U {kd}, ou k=1,2, •• ,p -1, a la puissance - l -p k < 3ph_l et reproduit A par somme.
6. APPLICATIONS
A L'ETUDE DES K-ARCS
Nous allons montrer enfin comment les resultats exposes dans les paragraphes precedents peuvent etre utilisee pour 1 'etude des k-arcs du plan PG(2,q).
Nous
premettons la definition suivante: Soit H un arc de PG(2,q); un point P, non situe sur l'arc, sera dit lie (par rapport
a H)
si au moins une corde de H passe par ce point; dans le cas contraire P
est dit Ubre.
686
G. Pellegrino
Un ensemble {P.} (i
~
1
bZe
a H,
1) de points, chacun libre par rapport a H, est dit agrega-
si {H u {Po }} est un arc. 1
Cela dit, prenons sur une conique C -que dorenavant nous supposons, sans alterer les generalites, representee par les (1.3), (1.4)- k < (q+l) pOints C et i disons H l'arc incomplet forme par ces points. Selon les notations et les symboles deja introduits, soit Z une droite par Y et soit nz (eventuellement
rj, rj,
si Zest tangente a C) le cycle du groupe n des Z
projectivites ayant l'axe sur Z. Nous noterons J = {r l ,r 2, •• ,r k } 1 'ensemble des places (indices) que les points de H occupent dans le cycle ~Z; J est un sous-ensemble de A={O,l, .• ,m-l} OU m = q_+1,p; en outre tout point P=P (eventuellement P=P .) (n n nJ n par la puissance w de la projectivite w generatrice de n •
m) est determine
~
z
A10rs i1 resulte de la discussion developpee dans les n.os 1,2,3,4 que l'on a: 6.1.
Pour chaque coupLe d 'eLements distincts r.,r. de J, teLs que r .+r .=n 1
J
1
J
(mod m), il existeune secante de H qui passe par Ze point P=P (=P .) de Z. n nJ La proposition 6.1 constitue un critere pour reconnaitre si un point Pest libre par
ra~port
a H. Ce critere sera constamment applique dans les constructions que
nous allons indiquer. 6.2. Soit C une coni que et Z une droite exterieure a C.
Notre but est de
determiner le nombre k de points qu'il faut prendre sur C afin que tout point de
Z soit lie par rapport au k-arc forme par les points pris sur c. Ayant pris sur L un point Y, exterieur a c, disons L', L" les points communs a c et a la polaire, x, de Y par rapport a c; disons en outre O,X deux points conjugues harmoniques par rapport au couple L' ,L".
On peut alors representer C
par la (1.3). Determinons le cycle ~z={sO'~l""~q} (si E C) du groupe 6 Z des projectivites axiales ayant 1 'axe sur Z. Prenons en 6 les points si dont les Z indices forment 1 'ensemble Z construit en 5.3 {Z c A = (O,l, •• ,q}); comme Z reproduit A par somme, tout point de l est lie par rapport au k-arc. k
On a donc
= IZI = d'+d"-l ou d' est un diviseur de q+l=d'd".
Par des modifications appropriees, cette construction est valable aussi dans le cas ou Lest secante ou tangente
a C.
Par la methode exposee, nous retrouvons 1a proposition (cf.[ 11], II, p.12)
Sur les
6.3.
k~rcs complets
687
de PG(2,q).q impair
(q+5)/2 points, arbitrairement pris sur une coni que C, determinent un
arc H par rapport auque1 est lie tout point du plan, sauf les points de {C -H}. D~MONSTRATION: C est representee par la (1.3).
It n HI
Soit tune droite par Y.
Si
2, tout point de t est lie; nous considererons donc le cas OU
Iz n HI 0,1. D'autre part, comme It n ci = {0,1,2}, nous avons a examiner les trois cas suivants. i) It n cl = O.
En ce cas Inzl = q+1 et 1es points de H sont un sous-
ensemble, de puissance (q+5)/2 de n • Si nous considerons 1es ensembles des t indices, on a m=q+l, tandis que 1es indices des points de H forment un sousensemble Z de puissance k=(q+5)/2 = (m+4)/2. Alors de 6.1 et 5.1 il resulte que tout point de Z est lie par rapport a H. On peut appliquer 1es memes conclusions aux autres cas en observant que: ii) pour It n cl = 1, on a I r,zl = q (impair); 1e nombre des points qui sont en r,z est (q+5)/2 si le point C = Z n C n'appartient pas a H; (q+3)/2 dans le cas contraire. Considerant les ensembles des indices, nous avons maintenant m+4
m = q; k = (q+3)/2 = [21 ; iii) pour It n cl = 2 on a I ~I = q-l; le nombre des points de H qui sont en ~
est au moins (q+3)/2, en accord avec 1 'hypothese sur Iz n HI.
Pour les
ensembles des indices on a m a q-l et k > (q+3)/2 = (m+4)/2.
7. CONSTRUCTION D'ARCS COMPLETS D'ORDRE k ~ (q+l)/2 Nous allons indiquer la construction en PG(2,q) d'arcs complets d'ordre k ~ (q+l)/2.
Ces constructions, comme nous avons deja dit, se referent a des
valeurs particulieres de q. 7.1.
Dans le plan PG(2,23) prenons sur la conique
C
(d'equation (1.3»
les
points suivants, dont nous avons indique -outre les coordonnees affines- meme les coordonnees parametriques dans la representation (1.4) C1 = (8,11) C2 = (8,12) C3 = (15,11 ) C4 = (15,12)
(5) (18 ) (14 ) (9)
C5 a (11,8) = (13) C6 = (11,15) = (10)
Al = (9,9) = (19) A2 = (9,14) a (14)
C7 = (12,8) = (16) C8 = (12,15) = (7)
A3 = (14,9) = (17) A4 = (14,14) = (16)
(7.1)
C. Pellegrino
688
Par rapport au 12-arc H = {C.,A.} (i=1, •• ,8;j=1 , •. ,4) sont lies tous les points 1
J
du plan exceptes les suivants: {C-H),
(1,~4,O),
(1,~6,O).
DEMONSTRATION: Nous rapportons entierement les comptes elabores, meme dans le but d'illustrer concretement le procede de construction. Des considerations developpees en 2.6, tenant compte des (2.14) et partant de
~l
= 21, nous pouvons construire le cycle
noo
du groupe
boo
relatif
a la
droite
z=O, (pour 1aque11e on a a=oo, 11=-1).
a
relatifs de
~1
De noo ' par la (2.18) on deduit les cycles toute autre droite passant par Y et exterieure a C. De meme, partant du groupe b O relatif a la droite x=O (pour on deduit 1es cycles re1atifs a toute droite par
= 14, on construit le cycle
laque11e est a=O, n=l). Y. secante groupes
a c.
De
9b
~
Dans notre cas, dans la (4.4) on a e = 1 et donc chacun des
Z'''Z"opere dans un seul cycle r'. r" qui est donne, respectivement, par (4.14), (4.25). La situation est resumee dans 1e tableau suivant. 6
689
Sur les arcs compiets de PG(2,q),q impair
/k
Ko
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 00 22 1 >100 00 21 5 4 7 20 22 15 10 6 14 11 o 12 9 17 13 8 1 3 16 19 18 2 00 19 10 8 14 17 21 7 20 12 5 22 0 1 18 11 3 16 2 6 9 15 13 14 17 19 2 ~7 ex
1T
7 14 3 >17 00 17 18 7 4 fi co 15 l8 5 10 6 >15 0011 16 5 8 >21 00 7
15 12 21 14 20 22 7 18 19 10 o 13 4 5 16 1 3 9 2 11 20 16 5 11 19 14 17 1 10 21
8 6
0 2 13 22 6 9 4 12 18 7 3 8
7 1 19 5 17 21 14 13 15 20 0 3 8 10 9 2 6 18 4 22 16 12 17 9 10 22 15 5 11
2 20 19 0 4 3 21 12 18 8 1 13 14 6 16
9 ~( 00 5 12 13 17 19 14 20 21 8 11 7 o 16 12 15 2 3 9 4 6 10 1 18 6 17 12 fl6 00 22 14 2 15 10 11 19 5 3 7 17 0 6 16 20 18 4 12 13 8 21 9 1 00 20 19 6 22 7 10 11 15 9 21 5 o 18 2 14 8 12 13 16 1 17 4 3 ~1 15 13 ~1
120 11
2 20 16 fl2 00 14 11 18 20 21 7 10 22 4 17 15 0 8 6 19 1 13 16 2 3 5 12 9 3 21 18 '3 fa 10 21 3 11 15 5 17 19 16 22 14 0 9 1 7 4 6 18 8 12 20 2 13
~O 1 0 ~2 00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
~1
22
r' 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 rIO F 1 12 8 6 14 4 10 3 18 7 21 2 16 5 20 13 19 9 17 15 11 22
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ~1 + ;; 0 1 1 fI fo 14 21 15 16 19 6 10 3 12 18 0 5 11 20 13 17 4 7 8 2 9 1 22 0 9 4 2 ~9 fa 5 19 7 9 15 12 20 6 1 13 o 10 22 17 3 11 8 14 16 4 8 2 21
~~
10 9 3
'\J 00 19
17 22 2 11 18 7 9 13 8
o 15
10 14 16 5 12 21
1 6 4 3 20
9 16 4 0g 0010 15 14 18 7 1 17 12 2 3 o 20 21 11 6 22 16 5 9 8 3 4 19 00 15 11 21 4 22 13 14 18 3 16 0 7 20 5 9 10 1 19 2 12 8 6 17 4 13 6 ~ 13 8 8 ~3 00 20 7 5 13 14 2 11 1 4 6 o 17 19 22 12 21 9 10 18 16 3 8 15 0011 5 20 6 10 8 21 4 16 1 o 22 7 19 2 15 13 17 3 18 2 9 14 15 12 9 ~5 7 22 19 8 21 3 5 13 6 9 o 14 17 10 18 20 2 15 4 1 6 12 11 4 6 12 fl4 00
1 8 13
~1
2 3 16
~
8 2 18
'b
1 17 9 15 16 18 4 o 19 5 7 8 14 6 22 12 3 2 13 10
00
21 20 11
00
17 14 10 3 5 4 22
2 8 12 o 11 15 21
'" 22 10 17 12 20 16 19 8 9 2
o 21
1 19 18 20 13 9 6 16 7
14 15 4 7 3 11
6 13 1 18 5
690
C. Pellegrino
Dans les cycles r', r" n'apparaissent pas les pOints 0 et '" respectivement; de marne, dans les cycles
~.
relatifs aux droites secantes
a c,
n'apparaissent pas
les points + /;: de c. -
1
Les droites d'equation
x=~S; x=~9; x=~11
contiennent les suivants couples de
points de c: (A,.A2).(A3,A4),(C"C2),(C3,C4),(CS,C6),(C7'CS); done tout point de ces droites est lie par rapport a H. De la table des cycles l'on deduit tout de suite cel1e des indices (places) que les pOints de
r~
assurnent dans 1es cycles relatifs aux droites passant par Y.
La table des indices est rapportee ci-dessous. A' cOte de chaque ligne sont rapportes les indices qui ne sont pas reproduits par somme par 1 'ensemble des indices que nous avons ecrits sur la
m~e
ligne. 3, 9, 15,
12 , 3 4 8 9 10 14 15 16 20 21 22 ! 117 1 2 4 S 7 10 14 17 19 20 22 23 1 5 8 9 10 11 13 14 lS 16 19 23 11
'"
1'7
I '18
3 4 6 7 8 10 14 16 17 18 20 21
, Q
2 4 5 6 8 9 15 16 18 19 20 22 1 2 \ 3 4 7 11 13 17 20 21 22 23 1 3 4 S 6 11 13 18 19 20 21 23
S
i '16 0
20
%
2 5 7 8 10 11
13 14 16 17 19 22
1'l21
2 3 S 6
13
11
15 18 19 21
r"
n0
1 4 5 6 7 10
Sl19
1 2 3 4 8
(/3 ! r'
! llo
I \113 Q
4
i
I
22
10 14 15 17 18 21 23 1 6 7 8 9 11 13 15 16 17 18 23 4 5 6 7 .9 10 13 14 16 17 18 19 4 5 6 7 9 10 13 14 16 17 18 19
tllz
t
9
21
1 3 6 7 9
1
2 6 7 8
12 15 16 17 18 21
10 12 14 18 19 20 21 9
13 14 15 16 20 21
2 3 4 S 9 10 12 13 117 18 19 20 1 3 7 8 9 10 12 13 14 15 19 21
Alors, par 6.1, soot libres par rapport
a H le
seu]s points P3. P9,P15,P21 de ]a
droite z=O (en correspondance de 1aque11e nous avons cycles, en n= on a libres par rapport
~3=4; ~9=6; ~15=17; ~21=19;
a H sont
(1,~4,0).(1,~6.0).
~=22).
Dans la table des
done, par la (2.15), les pOints Or i1 est tres facile de completer
Sur /es arcs comp/ets de PG(2,q),q impair
691
H en lui agregeant deux des points libres et, eventue11ement, d'autres points de
c. Les 12-ensemb1es de la table des indices (sauf
Qoo) sont des solutions partiel-
les du probleme 5.0 pour m=24,23,22. Maintenant demontrons le /'
,
THEOREME 6: En PG(2,23)
i~
existe un ara
aomp~et
d'ordre(q+l)/2=12.
/
DEMONSTRATION: Prenons sur C les points Ci (i e l, •• ,8) definis par la (7.1) et posons encore H = {C }. De 1a table des cycles on passe a celle des indices des i points C dans les differents cycles Q~. A cote de chaque ligne sont rapportes i les indices qui ne sont pas reproduits par somme par 1 'ensemble des indices ecrits sur la
m~me
ligne et aussi, compte tenu des (2.15),(4.13),(4.26), 1es coordonnees
des points du plan correspondant aux indices qui ne sont pas reproduits (c'est-adire les coordonnees des points du plan qui sont libres par rapport a H).
r-~!~ -T2[418FOll-4116-120-122~l7T9F~rl3ll5Tl-~1-9T21T23l(-1~~2~O),( 1 ,~3,0), (1 ,~4:0)-,( 1-,~6.0): (1-,;8-,O--)~-(~:;11-:;;j
I Si17'2:417il0jl4
17!20 221
I:
!
ili
4:
20:23;
J:
I i 9112 15 i i :i i ii i 3 4 7 10 14 17 20,21 : :, :2151819:12 1511619 221 ~8 221 % 258 9 15 16119 22 : i2 518 9 112 15116119 i I I 15 1 3 4 I 7 17 20 i21 i23 9 i12 I I i ~6 I I 6 11 I 1 3 6 11 13 18 21 23 13 118 23 1 lIzo 1 4 20 23 % 2 5 8 10 14 16 19 22 117
(17,~5).(17,~6)
5 18 9 ill 13 15 16 19
I
I
(7,~7),(7,0)
i 1
I
(16,~7),(16,0)
I I
(2,0)
5 7 9 10 13 14 16 18
r" II
13 1923
(3.~3),(3,~5).(3.~10)
10
13
(1.::.9 )
5 7 9 10 13 14 16 18
10
13
1 4 7 10 12 15 18 21
1 2 4 7 10
1119 1 3 4 10 12 18 19 21 678 9 13 14 15 16
10
0
~O
~ 134 5 17 18 19 21 ~4 6 7 8 10 12 14 15 16 1113 234 5 17 18 19 20 114 1 7 8 10 12 14 15 21
! I
(-1.~9)
12 15 18 20 21
(0.~2), (0.,:!:.5),
12
(19,::.4 )
4 10 11 12 18
(9.::.1), (9,0)
10 11 12
(14.,:!:.1 ),(14,0)
10
12
(0 .,:!:.7). (0 .,:!:.9). (0,,:!:.10)
(10,,:!:.3), (10,~5), (10.~0)
10 11 12
4 10 11 12 18
I
1
(6.~5).(6.~6)
12
r'
I
(20,~3),(20,~5),(20,~10)
f''2 1 3 6 7 17 18 21 23 1 6 11
I
!
(21,0 )
1 6 9 11 13 15 18 23
(18 ,~6), (18,~1O), (18,0)
·(5.~3),(5,~5),(5,~6),(5,~10),(5,0)
12
'3
!
(18,.:~:.3), (18,~5),
5 6 9 11 13 15 18 19
liz 1
i
(13 .~3), (13 .~5) , (13,0) (4.~4)
i
Sur les arcs complets de PG(2,q),q impair
693
Considerons les points M. = (+5,+5); on peut verifier -grace J
--
a la
symetrie
dans les coordonnees des points choisis- que tous les points de {C-H} et tous les points libres du plan sont distribues sur les 32 droites M.C .. A10rs si nous agregeons les points Mj
a l'arc
J 1
H, nous avons un 12-arc complet.
7.2. Le procede decrit en 7.1 a ete utilise pour 1es autres cas que nous avons examines.
Pour brievete nous rapportons seulement les resultats; nous nous
sommes refere toujours
a 1a
conique C d'equation (1.3).
i) En PG(2,29) i1 existe un arc comp1et d'ordre 14=(q-l)/2. On 1 'obtient prenant sur C les points
(~6,~9),(~9,~7+;(1,~12,O)
et, au dehors de C,
(~4,O),
(0,2:.4 ) • ii) En PG(2,31) il existe un arc comp1et d'ordre (~4,~4)
sur C et
(~10;0),(0,~10)
16=(q+1)2:(~7,~13),(~13,~7),
au dehors.
iii) En PG(2,37) il existe un arc complet d'ordre 18=(q-1)/2: (~18,~11)'(2:.8'2:.14),(~14'2:.8)
sur C et (-1,3,0),(-1,25,0) au dehors.
iv) En PG(2,41) il existe un arc complet d'ordre 18=(q-5)/2: (~16,~14)'(2:.12,~12),(l,~9,0)
sur C et
(~9,0+,(0,~9)
sur C et
(~5,O),{O'2:.5),{l'2:.15,O)
sur
C
et
(2:.9,~7),(~7'2:.9),
au dehors.
vi) En PG(2,47) il existe un arc complet d'ordre 22=(q-3)/2: (~22,~9),(~4,~19),(~19,~4),(~20,~20)
(~14,~16),
au dehors.
v) En PG(2,43) i1 existe un arc complet d'ordre 22=(q+l)/2: (~17,~20)'{2:.20,~17)
(~11,~18),
(1,~5,0)
(~9,~22),
au dehors.
BIBLIOGRAPHY 1. 2. 3. 4. 5. 6. 7.
A. Barlotti (1965). Some topics in finite geometrical structures. Institute of Statistic mimeo series n.439, Univ. of North Carolina. R.C. Bose (1947). Mathematiaal theory of the symmetrical faatorial design. Sankhya, 8, 107-166. C. Di Comite (1964). Intorno a certi (q+9)/2 arahi di S2,q, Atti Accad. Naz. Lince; Rend., 36, 818-824. D. Hughes and F. Piper (1974). Projective planes, Springer, Berlin. G. Korchmaros (1974). Osservazioni sui risuttati di B. Segre reZativi ai k-arahi aontenenti k-l punti di un/ovale. Atti Accad. Naz. Lincei, Rend. 56, 690-97. L. Lombardo Radice (1956). Sut problema dei k-archi aompZeti in S2,q' Boll. Un. Mat. Ita1., 11, 178-181. F.J. Mac Williams and N.J.A. Sloane (1977). The theory of error aorrecting aodes, North Holland, Amsterdan.
694
8. 9. 10.
1l. 12. 13.
14. 15. 16. 17. 18.
C. Pellegrino
G. Pellegrino (1977). Un'osservazione sul problema dei k-archi completi in 52' con q=l (mod 4), Atti Accad. Naz. Lincei Rend. 63, 39-44. G. Pe11eg~ino (1979). Alcune elementari proposizioni aritmetiche e loro applicazioni alZa teoria dei k-archi, Boll. Un. Mat. Ital. 16 A, 322-330. G. Pellegrino (1981). Archi compZeti di ordine (q+3)/2nei piani di Galois S2,Q con q=3 (mod 4). Rend. Circ. Mat. Pa1ermo, 30, 311-320. G. Pickert (1952). Projective Ebenen, Springer, Berlin. B. Qvist (1952). Some remarks concerning curves of second degree in a finite plane. Ann. Acad. Sci. Fenn. Ser. A, n.134. G. Scorza (1942). Gruppi astratti, Cremonese, Roma. B. Segre (1948). Lezioni di Geometria moderna, Zaniche11i, Bologna. B. Segre (1955). Ovals in finite projective plane, Canad. J. Math. 7, 414416. B. Segre (1959). Le geometrie di GaZois, Ann. Hat. pura App1., 48, 1-97. B. Segre (1960). Lectures on modern Geometry, Cremonese, Roma. F. Zirilli (1973). Su una eZasse di k-archi di un piano di Galois, Atti Accad. Naz. Lincei, Rend. 54, 393-397. Istituto Matematico Universita di Perugia Via Pascali 06100 Perugia Italy
Annals of Discrete Mathematics 18 (1983) 69S'{)98 © North-Holland Publishing Company
695
THE BUNDLE AXIOM AND EGGLIKE SUBSETS OF PROJECTIVE SPACES Nicolas Percsy
1. INTRODUCTI ON J. Kahn [9,10] has proved that Benz planes (i.e. inversive, Laguerre or Minkowski planes) satisfying the "bundle theorem" are egglike.
His result sug-
gests the following more general problem: find a class C of geometries (as large as possible), containing the Benz planes, such that a similar bundle condition can be stated for the geometries in C, condition which is sufficient for their embeddability in a three-dimensional projective space. We define such a class here, whose members are called egglike planar spaces; they generalize various well-known substructures of projective spaces such as affine spaces, semiquadrics, ovoids, cones, .... 2. EGGLIKE SUBSETS OF A PROJECTIVE SPACE Let P be a projective space of dimension n
~
2; the (n-l)- and (n-2)-dimen-
sional subspaces of P are called hyperpZanes and cotines respectively.
A subspace
S of P is called secant (resp. tangent) to a set Q of points of P is S n Q generates S in P (resp. S n Q does not generate S in P, but S n Q F ~). An egglike subset of P is a set Q of points such that any coline secant to Q is contained in at most one hyperplane tangent to Q.
The egglike subset is thick
if for any coline K, such that K n Q generates a hyperplane in K, the number of secant hyperplanes through K is not one. EXAMPLES: In a projective plane, an oval or a complete oval, the complement of a line (affine plane), the union of two lines, any set of pOints that meets no line in exactly one point are egglike subsets. PG(2,2) and PG(2,3).
They are all thick, provided ovals of
In a three-dimensional projective space, the complement of a
subspace, an ovoid, a cone (with oval or complete oval sections), an orthogonal or hermitian quadric are clearly egglike.
More generally, in arbitrary dimensional
N. Percsy
696
spaces, the complement of a subspace, a quadratic or semi-quadratic set in the sense of [3,5), the point-set of PG(5,2) representing the Witt design S(5,6,12) t12,13] are egglike subsets. Let us state a property, whose proof is easy, but which is helpful in checking the above examples.
For any point p of a projective space P, let us
denote by P the projective space whose points and lines are the lines and planes p
of P through p. PROPOSITION 1: Let P be a ppojective space of dimension eggUke if and only if fop each point p
E
~
3.
A subset Q of P is
Q, the secant lines thPough p constitute
an eggZike subset ill P . P
3. EGGLIKE PLANAR SPACES A lineap space (resp. a
p~anaP
space) means, as usually, a rank 3 (resp.
rank 4) geometric lattice or simple matroid (not necessarily finite); the elements (or flats) are called points and lines (resp. points, lines and planes) according to their rank.
For each point a
E
n of a planar space n, the lines and planes
through a are the points and lines of a linear space denoted by na' Let m be any transfinite cardinal number. oY'del'
I'l
if
for every point a
E
A planar space
TI
is egglike of
n, there is a triple (a,Q(a), P(a)), where Pta) is
a projective plane of order m, Q(a) is an egglike subset in it, and morphism (of linear spaces)from n onto Q(a). a Q(a) are.
a is
an iso-
The planar space n is thick if all
EXAMPLES: Any Benz plane B (defined in (7) and synthetically in [4)) has a natural structure of an egglike planar space n(B): the points, lines and planes of n(B) are the points of B,
the lines and pairs of points of B, the circles and certain
pairs of lines of B respectively; for each point a, the linear space na is an affine plane, together with one or two ideal points when B is a Laguerre or Minkowski plane; hence P(a) is just the projective plane obtained by adding all ideal points to na
= Q(a) and a is the identity. Similarly, inversive (Mobius),
Laguerre and 14inkowski planes in the broad sense (defined respectively in [1,2,6), also defined collectively and called Zykelebenen in (81) can be thought of as egglike planar spaces.
Locally projective planar spaces (in which na is a projec-
697
Bundle axiom and egglike subsets of projective spaces
tive plane for all points a) are egglike, as for instance the planar space associated to S(3,6,22). Let us remark that, by Proposition 1, a planar space isomorphic to an egglike subset of a 3-dimensional projective space (provided with its secant points, lines and planes) is egglike in the above sense.
4. THE BUNDLE PROPERTY
A bundle B of an egglike planar space IT is a family of planes such that for each point a
En
B, a(B) is the set of all lines of P(a), secant to Q(a), that
contain some pOint of P(a) (this point belongs to Q(a) if nB is a line in IT). The intersection nB is called the support of the bundle; it is clearly a line or a point of IT.
Two bundles B, B' are adJaaent - which is denoted by B-B' - if
- they have disjoint supports; - either their supports are both points, or they have a common plane.
We are
ready now to state a bundle property in egglike planar spaces, which coincides with the classical bundle property in the case of Benz planes or locally projective planar spaces [9,10] • Bundle property (B).
Given four distinct bundles, with pairwise disjoint
supports, if five of the six pairs of bundles are adjacent, then so is the sixth. If IT(Q) is the egglike planar space (of all secant points, lines and planes) of an egglike subset Q of a 3-dimensional projective space, then the bundles of IT(Q) are the families of all secant planes through a line which is secant or tangent to Q.
When Q is thick, it is easily checked that IT(Q) sat-
isfies the bundle property (B).
We think that the converse is true: any thick egg-
like planar space IT of order m satisfying (B) is isomorphic to an egglike subset of a 3-dimensional projective space of same order m (hence, it is embeddable).
We can
prove this with an additional restriction, which is a hypothesis of Kahn's result. THEOREM: Let IT be a thiak egglike planar spaae of (possibly infinite) order m; assume the following:
IT, there are n Zines of P(a) whose union aontains all . • h n .... .-- max {1 , Y -2m- 1 } when m -z-s . f·· po-z-nts of P( a ) Q() a , w-z-t -z-n-z-te and n f·· -z-n-z-te when m (#) For alZ points
is infinite.
a
E
698
N. Percsy 'i'hen
If
is "embeddable" in a 3-dimensional projective space of order m (as
the planar space of all secant points, lines and planes of a thick egglike subset).
The proof contains two steps: (1) by the embedding lenma [11, Theorem 4.3],
If
(which is locally embeddable) is
embedded in a locally projective planar lattice in the sense of Kahn [9,10]; (2) the latter lattice, satisfying Kahn's version of the bundle property, is then embeddable in a projective space by [10, Theorem 4] . The step (2) requires hypothesis (#).
We hope to extend Kahn's result by
dropping (#) in the case of egg1ike spaces and to obtain so a proof of the above theorem (without (#», which will be published elsewhere. BI BL IOGRAPHY 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
W. Benz, Uber Mobiusebenen. Ein Bericht, Jber. Deutsch, Math. Verein, 63 ( 1960), 1- 27. W. Benz and H. Maurer, Uber die Grund1agen der Laguerre-Geometrie. Ein Bericht, Jber. Deutsch. Math. Verein, 67 (1964:65), 14-42. F. Buekenhout, Ensembles quadratiques des espaces projectifs, Math. Z., 110 (1969) 306-318. F. Buekenhout, Les plans de Benz: une approche unifiee des plans de Mobius, Laguerre et Minkowski, to appear in J. Geometry. F. Buekenhout and C. Lefevre, Semi-quadratic sets in projective spaces, J. Geometry, 7 (1976), 17-42. K.J. Dienst, Hermitische Mengen vom Index 2 und ihre Bedeutung fUr Minkowski-Ebenen, Arch. Math., 33 (1979), 193-203. W. Heise and H. Seybold, Oas Existenzproblem der Mobius-, Laguerre und Minkowski-Erweiterungen endlicher affiner Ebenen, Bayer. Akad. Wiss. Math.-Natur. Kl. S.-B. 1975, 43-58. A. Herzer, BUsche1satze zur Charakterisierung projektiv darstellbarer Zykelebenen, Math. Z., 164 (1979), 215-238. J. Kahn, Locally projective-planar lattices ~hich satisfy the bundle-theore~ Thesis, Ohio State University 1979. J. Kahn, Locally projective-planar lattices which satisfy the bundle-theorem, to appear. N. Percsy, Locally embeddable geometries, Arch. Math., 37 (1981), 184-192. J.A. Todd, On the representation of the Mathieu groups as co11ineation groups, J. London Math. Soc., 34 (1959), 406-416. R. Wille, On incidence geometries of grade n, Atti Conv. Geom. Comb. Appl., Perugia 1971, 421-426. Universite de 1 'Etat a Mons avenue du Champ de Mars, 24 B - 7000 Mons Belgium
699
Annals of Discrete Mathematics 18 (1983) 699-712 North-Holland Publishing Company
KIRKMAN CUBES A. ROsa'" and S.A. Vanstone
1. INTRODUCTION Presently there appears to be quite a lot of interest in various generalizations of Room squares, such as Howell designs [1], generalized Room squares (3), Kirkman squares [6,11), generalized Howell designs [9], Room rectangles [4]- for a survey on these and various further generalizations of Room squares including those to higher dimensions see [10].
The concept of orthogonality of resolutions
of the associated underlying design is common to all these generalizations.
By
relaxing the usual concept of orthogonality, several additional interesting objects can be defined that still retain the esthetically and otherwise desirable properties of uniformity, regularity and balance.
In this paper we study the
existence of one kind of these objects in three dimensions, which we call Kirkman cubes [10] (as their underlying design is a Kirkman triple system).
2. BASIC DEFINITIONS AND PROPERTIES A uniform multidimensional generalized Room design of degree k, dimension d, multiplicity A and order v (briefly UMGRD (k,d,A,v), of [10]) is ad-dimensional array such that (i) every cell of the array is either empty or contains a k-subset of a v-set V, (ii) every element of V is contained in exactly one cell of any (d-l)-dimensional subarray (iii) every 2-subset of V is contained in exactly A cells of the array. Thus the set of k-subsets in the nonempty cells of a UMGRD ;s the set of blocks of a (v,k,A)-des;gn.
A UMGRD (k,d,A,v) is regular of index t (2
~
t
~
d)
if its projection on any t dimensions is a UMGRD (k,t,A,V) but projection on any (t-l) dimensions is never a UMGRD. Examples of UMGRDs include multidimensional
700
A. Rosa and S.A. Vanstone
Room designs (cf. [ 10J), or, for instance, a regular UMGRD (8,12,1,24) of index 2 obtained from the unique S(5,8,24) (see [4]). It is sometimes to our advantage to view UMGRDs as block designs possessing multiple resolutions.
To establish this equivalence, we need a few definitions.
If (V,B) is a (v,k,A)-BIBD then any set C of blocks in B that partitions V Any set of disjoint parallel classes R = {Cl' •.• ,C } r that partitions B is called a resolution; a BlBD admitting at least one resolution
is called a papallel class. is called resolvable.
Suppose now that a (v,k,A)-BIBD admits a set R = {R l ,R 2 , ••. ,R s } of s resoluiii i tions where R = {Cl, ••• ,C n }, C.'s are parallel classes. We call the set R a dJ i ol'thogonql set of resolutions if for any d parallel classes c. l , 1k
(where C.
Jk
E
Jl
k
R ).
A d-orthogonal set R of resolutions is said to be regulap of index t (2 < t < d) if R is a t-orthogonal set but contains no (t-l)-orthogonal subset.
The aforementi oned equi va 1ence is then almost immedi a te (cf. [10]): A regular UMGRD (k,d,A,v) of index t exists if and only if there exists a dorthogonal set of d resolutions of a (v,k,A)-BIBD that is of index t. In what follows we will be concerned with a special case of UMGRDs when k = 3, d = 3, A = 1 and index t = 3; these will be henceforth called Kirkman cubes and denoted by KC 3(v). [In contrast, a UMGRD with k = 3, d = 3, A = 1 and index t = 2 will be called a strong Kirkman cube SKC (v); until very recently, no exam2 ples of strong Kirkman cubes were known, however, just prior to this meeting, the second author succeeded in constructing an SKC (255).]. 3 Thus, a Kirkman cube KC 3(v) is a 3-dimensional array on v elements such that the set of triples in the nonempty cells of each of its planes is a parallel class, and the set of triples in the nonempty cells of the whole array is the set of triples of an STS(v) while the projection on any two dimensions never yields a Kirkman square (i.e. a
Ur~GRD
with k = 3, d = 2, A = 1, t = 2, cf. [2]).
Kirkman cubes
701
3. KIRKMAN CUBES OF SMALL ORDERS Obviously the trivial necessary condition for the existence of a KC (v) is 3 that v 3 (mod 6), v> 3. Since the affine plane of order 3 has a unique resolu-
=
tion there exists no KC 3(9). It is well-known that there are four resolvable Steiner triple systems of order 15. Of these, No. 1.1 on the list of White, Cole and Cummings [12] (= PG(3,2)), as well as No. 1.7 on the same list, admit a set (actually, several sets) of three 3-orthogonal resolutions, while the remaining two resolvable STS(15) do not admit such a set.
Thus a KC (15) exists, and an 3 example of such is in Table 1. There also exist several KC (2l)'s (see [5]), and 3 an example of a KC 3(27) is easy to obtain from AG(3,3) (see Table 2.) The existence of a KC 3(33) is undecided at this point. A general method of constructing a KC (v) directly is the well-known 3 starter-adder method (see, e.g., [3]). The smallest order for which we succeeded in constructing a Kirkman cube by this method is v = 39 (see Table 3). Potentially important for building up small order Kirkman cubes is a "triplication method" producing a KC 3(v) from a KC 3(v). Unlike in the case of Kirkman squares, this method is at least feasible and has indeed been used to obtain a KC (45) and a KC 3(63). The construction involves, as is to be expected, a 3 "twisted" direct product. Unfortunately, the general features of one of the components of this construction remain elusive so that at present at least the triplication is only a hypothetical construction in general (although it has been successful on every order on which it has been tested).
In view of the results of
the next section, this may be all that is needed to completely settle the existence question for Kirkman cubes.
4. MAIN RECURSIVE CONSTRUCTIONS In this section, we describe various recursive constructions for KC 3 (v)s. The constructions are, of course, more general and will apply to any Ur~GRD (k,t,A,v).
The most powerful of these constructions is PBD-closure
(see Wilson
[ 13] ).
Let B be a set of k elements. Define VG = B x from the elements in B x {1,2}. the symbol set VB'
{l
,2} u
{oo}
where
00
is distinct
Suppose there exists a KC3(2IBI + 1) defined on
Coordinatize this cube KB by the elements of B.
Let
KB(x,y,z) be the block of K contained in the cell (x,y,z);of course, KB(x,y,z)
=~
702
A. Rosa and S.A. Vanstone
if the cell (x,y,z) is empty.
Without loss of generality, we assume that
KS(x,x,x) is the block {oo,xl,X } of K. Z place of (x,i).). THEOREM 4.1:
(NOTE: For convenience we write xi in
D be a PSD(v;K) such that for each k E K there exists a
La
KC (Zk+ 1 ) • 'rhen thei'e exists a KC 3 (2v + 1). 3 PROOF: Let V be the point set of D and 6 the block set. where
00
if. V x {1,2}.
Coordinatize a v
x
v
x
;'t
Define V
=
V x {1,Z}U
roo}
v cube K'" with the elements of V.
;.:
For each S E B, consider the subcube of K determined by the elements of B.
The
entries (cells) in this subcube are filled in by the entries in KB• We now show thatK isaKC 3(2v+l). Cons i der any pa i r {a., b. } c /'. J
1
in a unique block B of B. :'c
unique cell of K. a
-
If
j,
then a 1 band {a,b} is contained
Then ai' bj is in a unique cell of KB and, hence, a
If i 1 j and a 1 b, the same argument holds.
If i 1
j
and
= b then {a.,b.} is contained in cell (a,a,a) of K*. It is easily seen that 1;:
J
}, a E V -{oo} , is contained in a unique cell. i :', cells of K are the blocks of a (2v+l,3,l)-BIBD. {~"ai
Therefore, the nonempty
Consider any plane of K*.
For instance, the cells (a,x,y), for all x,y E V,
form a plane of K'" for fixed a.
Let Bl ,B , ••• ,B be the blocks of D which contain 2 t
a.
Then, {B i - {a}: 1 .;; i .;;
t}
is a partition of V-fa I. ,,
k> 6, then q < (k - 4)4 /4 • h-s 2) If k = u2 ,u odd, 1 < u < 2s-2 ,then q .,;; 4 (u-2) 2(2 s - u - 2) 2.
1) If k
Sia F un campo finito; un polinomio, P, appartenente al dominic di integrita FIx) dei polinomi nell'indeterminata x a coefficienti in F si dice di permutazione su F se t
~
P(t)
e una
corrispondenza biunivoca su F.
Sia d'ora in poi F un campo finito d'ordine q q
= 2h, q> 2 e siano A(q) e
P( q) rispettivamente il piano affine su F e il completamento proiettivo di A(q). q
Se P E FqI x), indicheremo con D(P) il seguente sottoinsieme di P(q) D(P) = {(P(t) , t) ; t
E
Fq } U {Xco , Yco }
(X"" Y", punti all'infinito degli assi). Sotto opportune ipotesi per P, D(P)
e un'ovale
ed
e nota
che ogni ovale di
P(q) e proiettivamente equivalente ad un'ovale D(P) 12, 3) • Se P(t) = t k porremo D(k) = D(P) e chiameremo monomia una tale ovale.
Sup-
posta k < q (caso al quale ci si puo sempre ricondurre), si ha che D(k) e un'ovale monomia se e solo se sana soddisfatte le seguenti condizioni 1) k e pari, 2) (k,q-l)
= 1,
3) (k-l , q-l) = 1, 4) (x k + l)/(x + 1)
e un
polinomio di sostituzione.
Sono note tutte le ovali monomie di P (2h) con h " 8 (1). In questa nota si dimostra che, se D(k) e un'ovale monomia di P (2 h ), si hanno i seguenti risultati
714
L.A. Rosati h
=6
h
2
2
(mod 8), 2 > k> 6, al10ra 2 < (k - 10k + 20) /4, hs s k 2 s 2 2) se k = u2 - , u dispari, 1 < u < 2 - 2, al10ra 2 ~ 4(u-2) (2 -u-2) . 1) se k
1.
Sia
~
= 6(k) 1a curva di P(q) di equazione (in coordinate affini)
(l
+ l)(y + 1) + (x+l)(/ + 1)
0.
=
(1)
= 0, x + Y =0 e in un'altra componente, che chiameremo r = r(k), 1a cui equazione e 6
si spezza ne11e tre rette di equazioni rispettive x + 1 = 0, y + 1 2
1 + x +y + x
LEHt~
1:
S~a
2 k-2 k-2 + xy + y + ••• + x + ••• + Y = 0.
k = 6 (mod 8), k > 6; aHara r
,:: .oue gener'e si ha g
e assolutamente
(2)
iniducibile.
Se 9
e
~ (k-4)2/ 4•
DIMOSTRAZIONE: Poiche k
1
e dispari, le intersezioni di r con la retta impropria
sono distinte, cosicche r non ha singolarita all 'infinito.
Detto f(x,y) il primo
membro den' equazi one (1), s i ha ?,f/ax
= y
Percio, se (a,S) ,~
k
k = B
k
+ 1,
af/ay
=
e un punto singolare di
k
x + 1.
6,
si ha
=1
(3)
e quindi a k/ 2 = sk/2 = 1.
(4)
Poniamo x = X + a, y = Y + B, F(X,Y) = f(X + a, Y + B).
e k =6 (mod 8), a 4 e data da Ct
k-2
Tenuto conto che
risulta che la somma dei termini di F(x,y) di grado non superiore
2 k-2 ? k-2 2 k-2 2 k-4 4 k-4 4 (B+l)x +B (a+1 )y1-a x y+B xy +a (l+B)x +8 (l+a)y.
(5)
Pertanto, detta 6(a,8) la curva di equazione F(x,y} = 0, si ha che (O,O) un punto triplo per 6(1,1); corrispondentemente (1,1) lora non appartiene a
e un
punto triplo di
6
e
e al-
f.
Ana10gamente si vede che, se ", e quindi punti semplici di r.
a F
1, a
k
1,
(a,l),
(l,a) sono punti doppi di
Sulle ovali di PG(2,q),q pari
k k Se a = e = 1, a ~ 1, S ~ 1, (0,0) a = B. con
Sia a
a
k-2
~
f B. un punta semplice se
B; la tangente cuspidale, m, ha un contatto almeno quadripunto
in (0,0)
~(a,B)
e un punto doppio cuspidale di ~(a,S).
e un punto doppio cuspidale di r se a
Pertanto (a,S)
715
fa parte di
0
~(a,B)
se e solo se il sistema
2 k-2 2 k-2 k-2 (S+l)x +8 (a+l)y =0, a x+8 y=O
ha una soluzione non nulla, ossia se e solo se
8+
a
a +
saO.
(6)
In questa ipotesi il sistema formato dalle due equazioni precedenti e dal1 'equazione a
k-4
4 k-4 4 (l+S)x + S (l+a)y =
°
ha sol tanto la soluzione nulla, ossia m ha un contatto solo quadripunto con Corrispondentemente (0,0) grado in
e un tacnodo di
~(a,8);
~(a,8).
infatti l'equazione di secondo
A
-2 -4 -2 2 4 -8 a A + a (1+8) + 8 A + 8 a (l+a) = ha due radici distinte e diverse da
° e dalla
° (5)
si ottiene che, se A e una qua-
lunque di esse, allora la parabola di equazione y
= 82/a 2 x +
e osculatrice
~(a,S)
AX
2
in (0,0).
Riepilogando, se poniamo k = 4t + 2, tenuto con to della (4), si ha che r possiede esattamente 4t 2 - 2t punti singolari; ciascuno di essi e un punta doppio cuspidale.
Detto m il numero dei tacnodi, per la (6) si ha m ~ 2t; i rimanenti
punti sono cuspidi di prima specie.
r non contiene rette.
r e irriducibile. Supponiamo infatti che r si spezzi sulla chiusura algebrica di Fq in due componenti irriducibili, fl e f2' d'ordini rispettivi
r,s(r~s).
Ognuna delle intersezioni di fl con f2 e per f un punto singolare, che pero non puo essere una cuspide di prima specie; pertanto fl e f2 han no al massimo 2t
inte~
sezioni. Sia P uno di tali punti; P e semplice sia per fl che per f2' la tangente in esso a f e tangente a fl e a f2 e, poiche P e un tacnodo, P assorbe due delle intersezioni di fl con f 2• Di conseguenza, tenuto conto del teorema di Bezout, si ha r +s
4t,
rs
~
4t,
(7)
L.A. Rosati
716
dove rs
e pari e pertanto r,s sana entrambi pari. Dalla (7) segue
l/r + 1/5 ;;. 1.
Poiche r:6. Siccome r ha m tacnodi, 4t 2 - 2t - m cuspidi di prima specie e non ha altre s~2
singolarita, per i1 genere. g, di r si ha 9 .;;
2 4t - 4t + 1 = (k - 4)2/4.
TEOREr-1A 1: Sia O(k) (mod 8). k F 6.
Wl
'ovate monomia di P(q) (q
2h, h > 1) e sia k < q, k
="
6
A lZO?'a
q < (k - 4)4/4.
OHIOSTRAZIONE: Per assurdo supponiamo q: q - 2gvq;;.
Po;che (k - 1, q - 1)
Vq
(vq - (k - 4)2 12 )
;;. O.
= 1, ciascuno dei suddetti punti appartiene ad A(q) e sicco-
= 1 nessuno di ess; appartiene al1a retta y = x. Questo contro 1 'ik potes; che D(k) sia un'ovale. Infatti, se O(k) e un'ovale, (x + l)/(x+l) e un
me (k. q - 1)
k polinomio di permutazione, ossia (x +1)/(x+l)=(yk+1)/(y+1) solo se x = y.
Questa
contraddizione dimostra il teorema.
2.
Siano u, s due interi positivi, i1 primo dei qua1i dispari, tali che si abbia u < 2s e sia ~ la curva di P(2 h) (h> 1) di equazione (in coordinate affini) u
(x + 1) (y ~
5
2
+ 1) + (x
s 2
+ 1)
(i
+ 1) = O.
(8)
si spezza nelle tre rette di equazioni rispettive x + 1 = 0, y + 1 = 0, x + y =0
e in un'a1tra componente che chiameremo r. LHII4A 2: Se u F 1,25 - 1, s (2 - u - 2).
E
e assolutamente
il'I'iduaibil-e e il. suo genere
e
(u - 2)
Sulle ovali di PG(2,q),q pari
DIMOSTRAZIONE:
~
Le tangenti a ~
717
ha due punti di molteplicita u rispettivamente in Xoo e in Yoo ' in ciascuno di essi sana distinte ed hanno l'equazione comp1essiva
rispettivamente yU + 1 =
°e
XU +
Le a1tre intersezioni di ~ con 1a retta
1 = O.
impropria sana date da11'equazione y
2s _u
+x
2s _u
= O.
(9)
Sono dunque distinte (2 s _u
e percio ~ non ha punt; singo1ari al-
e dispari)
l 'infinito diversi da X e da Y • 00
00
Detto f i1 primo membro dell 'equazione (8), si ha 2s u-l 2s u-1 af/ax = (y + l)x, af/ay = (x + l)y , e quindi gli untci punti singo1ari a1 finito sana 0=(0,0) e, se u t 1, U = (1,1). In particolare si ha che 1e rette x + 1 semplici di
= 0, Y + 1 = 0, x + y = 0 sana componenti
~.
° e un punta di molteplicita U;
. · l ' equazlone comp1 eSSlva xu + yu
Posto x
=X+
1, y
=Y+
le tangenti in esso sono distinte e hanno
= O.
1, si ha
s s ((X + l)u + 1)y2 + X2 ((Y + l)u + 1)
s e pertanto U e un punto di molteplicita 2
+
=0 1; 1e tangenti in U so no distinte ed
hanno l'equazione complessiva (x +
1) (y
+
1)
((x +
2
s
1) -
1
s
+ (y +
1)
2 -1
) = 0.
Di conseguenza i punti singolari di r sono Xoo ' Yoo ' 0, U e le mo1tep1icita rispettive sono u - 1, u - 1, u - 1, 2s - 2. Le tangenti a r in c;ascuno di questi punti sono distinte. Sia a la trasformaz;one quadratica di P(2 h) in se avente per punti fondamentali Xoo ' Yoo ' U; corrispondentemente siano X', Y', U' i punti fondamenta1i di a a muta r in una curva, r', d'ordine 2s - 2 non passante ne per X', ne per Y' e s avente in U' un punto multiplo a tangenti distinte di molteplicita 2 - u - 1.
-1
r' ha in a(O) = 0' un punto mu1tiplo a tangenti distinte della stessa moltep1icita u - 1 di O.
Dunque le uniche singo1arita di r' so no 0' e U'.
Supponiamo ora r riducibi1e su11a chi usura a1gebrica, r'
e riducibile;
F,
di F.
Su
F anche
siano r , L2 due componenti irriducibi1i distinte di r' e siano l r 1,r 2 gli ordini rispettivi.
718
L.A. Rosati
Poiehe I' non ha singo1arita diverse da 0' e da U', 1e intersezioni di L1 con Z2 si trovano fra questi due punti e pereio, se indiehiamo con m., n. 1
(i = " 2) 1e mo'tep1ieita di
~i
1
rispettivamente in 0' e in U', per i' teorema di
Bezout s i ha r,r 2
m1m2 + n1n2 •
('O)
0' altronde r.;;'u.+n.(i 1
1
1,2}
1
e di conseguenza m,m 2 + n1n2 ;;. (m 1 + n,}(m 2
+
n ) 2
n,m 2 = 0
m,n 2
e quindi deve verifiearsi una de1'e seguenti quattro eondizioni
= n1 = 0, m2 = n2 = 0, m, = m2 = 0, n,
m,
n2
=0
Per 1a (10) s i hanno a 11 ora due possibilita a)
m 1
m 2
0,
r
1
n"
r2
n , 2
b)
n 1
n2
0,
r1
m, 1
r
m, 2
e pereio
I.
1
(i
2
= 1,2) e una retta (poiche e irriducibi1e, ha ordine r.1 e un punto
di mo1tep1icita r i ) e 1e due rette [1' [2 0 passano entrambe per 0' e non passano per U' 0 viceversa. Questo signifiea che [' e costituita da rette passanti 0 tutte per 0'
Tenuto conto che le mo1tep1ieita di [' in 0' e in U' no rispettivamente u - 1 e 2s - u - 1, si ha 0 u = lou = 2s - 1. 0
tutte per U'.
s~
11 resto dell 'enunciato segue subito. TEOREMA 2: Siano u, h, s tre interi positivi soddisfacenti alle seguenti condizioa) u
e d'[spw'i,
b) 1
2d.
(14 )
-
U - 2) ~ 2 e, poiche d ~ h/2,
720
2) h
L.A. Rosati
e dispari. Si ha M;;' 2h/2(V2 - 1)
+ 1.
Ora 2h/2(V2 - 1) ;;, 2h/ 3 per h;;' 9 e, poiche d ~ h/3, 2h/2(V2 - 1) + 1> 2d
(15 )
per h ;;. 9. Siccome per 1e condizioni b) e c) de11'enunciato si ha h> 3 e poiche 1a (15)
e vera anche per h = 5, 7, anche in questo caso si ha M> 2d e al10ra dal1a
(14) segue N> O.
Questa contraddizione dimostra i1 teorema.
BIBLIOGRAFIA 1. 2.
3. 4.
J.W.P. Hirschfeld, Projective geometries over finite
fie~ds, Oxford, 1979. B. Segre, Sui k-archi nei piani finiti di caratteristica due, Rev. Math. PI/res AppZ., 2 (1957), 289-300. B. Segre, Ova1i e curve 0 nei piani di Galois di caratteristica due, Atti Accad. :'iaz. i-incei Rend., 32 (1962), 783-790.
A. We i 1. Sur
~es
courbes a 1gebriques e t les varie tes qui s' en deduisent,
Hermann, 1948.
Istituto Matematico "U. Dini" Universita degli Stud; di Firenze Via1e Morgagni 67/A 50134 Fi renze Italy
721
Annals of Discrete Mathematics 18 (1983) 721-724 North-Holland Publishing Company
GROUPS ON POINTS AND SETS Jan Saxl
In this note we wish to discuss three related problems concerning permutati on groups. 1. GROUPS ON UNORDERED SUBSETS Let G be a permutation group on a set n of v points.
o~
i
~
n
v/2 we write (i) for the G-space of all i-subsets of n with the induced
natural G-action. i
~
j
~
For any integer i with
Let n. be the number of G-orbits on (?). 1
v/2, as was proved by Livingstone and Wagner in (9).
Then n. ~ n. whenever 1
J
Our first problem
was already considered in (6): QUESTION 1: When is ni The case i detail in (6).
= ni +l > 1 ?
= 1 is very easy. We studied the next case, i = 2, in some The problem for i = 3 is much more difficult; Cameron investigated
the infinite case in (4), but the finite case remains to be done. also considered the situation where ni G.
Cameron (3)
ni+l = ni+2 > 1 for some infinite group
However, despite the considerable amount of attention it has received over the last five years, the problem remains wide open.
2. ONE POINTS AND BLOCKS OF DESIGNS Let V be a non-trivial 2-design, let G be a subgroup of the automorphism group of V.
Then the G-action on points induces a G-action on the set of blocks.
If G has m orbits on the set of points and n orbits on the set of blocks then it is well known that m ~ n (and there is also a more general version of this for 2s-designs).
~Saxl
722
QUESTION 2: When is m
n> 1 ?
Certainly m = n whenever V is a symmetric design, but if V is not symmetric then one would expect to obtain some information about V and G. tigated the situation when V is a Steiner triple system S(v).
In [9] we invesThe starting point
is the following simple observation: Given a pair of point-orbits choose a point in each; these determine a block and hence give a block-orbit.
It follows that
m(m-l) < k(k-l)n, where k is the block size (so k is 3 in our case). and k = 3 imply that m < 7.
Thus m = n
In fact, we show in [9] that m < 3, and obtain some
more information about the cases m = 3 and m = 2 If m = 3 then the point-orbits have sizes 1, (v-1)/2,(v-1)/2 and one of them carries a block-transitive hyperplane; there are many examples of such systems, geometrical and others. If m = 2 then one of a) the group G fixes a point and is transitive both on the set of lines on the fixed point and on the set of lines off the fixed point, or b) the G-orbits on points have sizes (v-1)/2 and (v+l)/2, with the shorter carrying a block-transitive hyperplane. We conjecture that in case a) the full automorphism group larger than G and is transitive on the points of V; it then follows that V is known by a theorem of M.
Hall [7].
We have not been able to prove this though.
We have, however, succeeded in classifying the designs in case b) - these are in general the projective geometries PG(d,2) over the field of 2 elements, the only exceptions occurring when v is 15 or 63. As far as I know, no work has been done on Question 2 for designs other than Steiner triple systems (as above) and projective geometries (see below).
3. ON SUBSPACES Let G be a subgroup of GL(V), where V is a d-dimensional vector space V(d,q) For i < d/2 let m be the number of G-orbits i in the action on the set of i-dimensional subspaces of V. Again, i < j < d/2
over a finite field of q elements. implies that m i
~
mj .
QUESTION 3: When is mi
mi+l > 1 ?
723
Groups on points and sets
As far as I know, no work at all has been done so far on the case where
> 1, so I shall concentrate on the case i = 1. Here one hopes to prove the following CONJECTURE: If G < GL(d,q) with d > 3 and with ml = m2 > 1 then G is reducible. In particular, G stabilizes a point, a hyperpZane or a non-incident point-hyperplane pair.
As a consequence of the work on Steiner triple systems mentioned above we see that the conjecture is true for q = 2. More recently, Cameron and Liebler [5] considered the conjecture in general and reduced it to some interesting geometrical problems in 3-dimensional projective spaces. 2
Their methods are sufficient to
handle the case where q + q + 1 is a prime number. considered the case where ml
Kantor [8) has previously
= m2 = 2.
REMARKS: 1. A general technique for proving inequalities like the ones above is described in [2). 2. The question in paragraph 3 can be adapted to subgroups of other classical groups. No work has been done on this yet. 3. The problems in part 1 were first studied in [6] as a way of investigating permutation groups with the interchange property (i .e. groups of even degree 2k which interchange any k-subset of the domain with its complement). It follows from the recent work of Babai [1] on the orders of 2-transitive groups that the interchange property problem has been reduced to a finite problem now.
BIBLIOGRAPHY 1. 2. 3. 4.
L. Babai, On the order of doubly transitive permutation groups. Invent. Math.~ 65 (1982). 473-484. P.J. Cameron. Permutation groups on unordered sets. in Higher Combinatories (ed. M. Aigner). D. Heidel Publ. 1977. pp. 217-239. P.J. Cameron. Orbits of permutation groups on unordered sets. J. London Math. Soe.~ 17 (1978), 410-414. P.J. Cameron. Orbits of permutation groups on unordered sets. II, J. London Math. Soe.~ 23 (1981). 249-264.
724
5. 6. 7. 8. 9. 10.
J. Saxl
P.J. Cameron and R.A. Liebler, Tactical decomposition and orbits of projective groups, to appear in Linear A~gebra Appl. P.J. Cameron, P.t1. Neumann and J. Saxl, An interchange property for finite permutation groups, Bu~l. London Math. Sao., 11 (1979), 161-169. 14. Han, Jr., Group theory and block designs, Proc. Internat. Conf. Theory of Groups Canberra 1965, Gordon and Breach Sc. Publ. 1967, pp. 115144. W.t4. Kantor, Private communication. D. Livingstone and A. Wagner, Transitivity of finite permutation groups on unordered sets, Math. Z., 90 (1965), 393-403. J. Saxl, On points and triples of Steiner triple systems, Aroh. Math., 36 (1981), 558-564.
Department of Pure Mathematics and Hathematical Statistics 16 Mill Lane Cambridge CB2 1SB, U.K.
Annals of Discrete Mathematics 18 (1983) 725-730 North-Holland Publishing Company
725
AN APPLICATION OF COMBINATORICS OF YOUNG TABLEAUX TO GEOMETRIC PROBLEMS Elisabetta Strickland
The purpose of this brief exposition is to give an idea of how the combinatorial devices introduced by Alfred Young with the name of "diagrams" and "tableaux" can be used in order to solve geometric problems, according to the results proved in [11. First let us see what are the questions one can give an answer to.
Let R be
any commutative ring with identity, V and U finite free R-modules, rank V = n, rank U = m.
In the affine space A = Hom (U,V) x Hom (V,U) consider the variety W
of pairs ('Pl''P ) of maps 2 U--+ V
'Pl : V ---+ U, 'P 2 such that 'Pl Fixed bases B(V)
o
'P 2
= 0U' 'P 2
0
'Pl = 0v
= {al •••• an } and B(U) = {bl, .•• ,bm} for each module, we can
identify Wwith the variety of pairs of matrices (Y,X) with entries in R, X being a mxn matrix, Y an nxm matrix, such that XY
= 0mxm
,YX
= 0nxn
Let W(k ,k ) be the subvariety of Wconsisting of pairs of matrices M ,M l 2 l 2 such that rank r~.1 " k., where k ,k are such that k . .;;; min (n,m). In [11 the l 1 1 2
equations of such varieties are given, in the following way.
Let
~
be the
coordinate ring of the affine space A, i.e. the polynomial ring R( X•. ,Y .. J, lJ Jl l ••• m, j = l •••• n and B = R[W1 the reduced coordinate ring of Wand let I be the ideal in A generated by the elements i,j = 1 .... n
and
E, Strickland
726
m z: y
k=l
ilt
X
i ,j
kj'
=1
•••• m
If 1(K ,k } is the ideal generated by and the determinants of the minors 1 2 of X (resp. Y) of size k1+1 (resp. kZ + 1) and B(k1,k Z} the reduced coordinate ring of W(k ,k ), in [11 it is proved that B(kl'k 2) ? Ii. / I(k 1,k 2 ). l 2 Moreover it is shown that, when R is Cohen-Macaulay and normal, and
kl+k2
~
min (n,m). W(k"K2l is Cohen Macaulay and normal too.
So let us introduce the combinatorial tools. A Young diagram
with k rows is a non i ncrea 5 i ng sequence of pas iti ve
0
integers
One can think of
a
as a sequence of rows of "boxes" of length °1
ThuS
I I 6,4,1
A Young tableau is a filling of a Young diagram a with integers out of 1,2 ..... is called the "shape" of the Young tableau. Given a Young tableau
T
where the a, ,'s are indices out of 1.2 ••• and one assumes lJ
m1 ~ m2 ~ •••• > ms ' then T is called standard if furthermore one has (i)
(ii)
(iii)
a"1J < a'k' when k > j, 1 a"
< a k " when k lJ J
~
i.e. the rows are strictly increasing sequences;
i, i.e. the columns are non decreasing sequences.
Young tableaux are useful tools because they are related to monomials in the
An application of combinatorics of Young tableaux
727
minors and representations. As a matter of fact we notice that a minor of a matrix X can be denoted by a pair of strictly increasing sequences. the row indices and the column indices. one of which can be written backwards. thus in the following way:
with 1
a < .... < a k ~ nand 1 ~ b < .... < b ~ n. l k l One can partially order the minors by declaring ~
I
[ak ...... a l iff k ~ hand a.
~ a~ 1
1
I
bl ...... bkl .;;; [ah ...... aj • b.
1
~ b~ 1
for 1
~
i
~
bj ...... bhl
h.
A standard monomial m = m m2 •••• mh in the minors m .m ••••• m is a l l 2 h . 1·1n wh·lC h m .;;;..:: ..:: mho monomla m ~ .... ~ 2 l A standard monomial in the minors of a matrix X is indicated by a pair of standard tableaux. which we write as a double tableau, i.e. a
1
a
1
°1
l
a2
2
a
°2
l
l b
l b
1
°1
2 b 1
b2
°2
aot .•• t represents the product of minors [a
1
°1
1
••••• a l
I
1
1
2
°
°
2
bl ..... b ][ a ..... al 1 2
2 2 t t bl •••••• b 1 ••• [ a ••••• a l °2 °t
t
t
bl ..... b
1.
°t
A good survey on the combinatorial properties of Young tableaux can be found in [2 J. Now on W(k l .k 2} we have a natural action of the group G = Gl(V) x Gl(U) defined as follows. Given (M l .M 2) E W(k l .k 2). (go.gl) E G. we define (+)
(go·9 1 ) (tl1·1~2) = (goMl gl
-1
• gl M2 go
-1
).
It is clear that (+) also belongs to W(k .k 2}. so the action of G on Winl duces an action of G on the reduced coordinate ring B(k 1 .k 2} of the variety W(k .k 2}. We consider B(k 1 ,k 2) as a representation of G. One can construct a 1
728
E. Strickland
basis of "srecial standard double tableaux" for B(k 1k2 ), therefore intimately connected with the action of G. Let us see what are the standard double tableaux in our case. Suppose the symbol (,.,)
denotes the determinant of the minor of the matrix X = (X st ) (resp. of the matrix
Y = (Y st )) if k = n (if k = m). whose rows are those of indices il •••• i s and whose columns are those whose set of indices is the complement {h < .... < hs } taken in l order. in {1 •••• n} [resp. {l ••• m}) of the set of indices J= d l ...• 3 } n-s [resp. j
dl
=
••••
times (-1)\ t being the sign of the permutation
hl ··•· hs ) [resp. (31 •••• 3m-s hl .••• hs )]. One can write the symbol (*) in the following compact expression:
(31 ••••
3n-s
3m-s })
(I
J] X • ( I
J] Y
(il •.•. i s ) and j = (3 1
where
J = (3 1 A =
I
••••
jn-s) in the first case and
3m-s ) in
P'l ..... 'A
the second case. Moreover. if M= (ml •••• ,m ) and r ), then we denote the determinant of a minor where rows (resp.
q columns) are those of indices (ml, •••• ,m r , Al' •.•• ,A q ) in the given order by (resp. [I] C (M
A)] ).
U
Consider now the tableaux: i lh
...
i 2h i
= (H Y I I
K' ) Y
(H
IK
)y
i
where 1
~
sl
...
i sh
jln-h
2
s
1
jlm-h
.....
1
j2m-h
~
jrt
~
We can associate to (H K)X' (H' K'}y polynomials in the coordinate ring 1J
J1
s
2
............ jsl ........ jsm-h
i rt < m and 1 ~ jrt < n in (1) and 1 ~ irt < nand 1
R[ X.. ,Y .. J,
2
jsn-h
jll j21
1
j2n-h
jsl
s
i 2h
i 21
jll j21
2
i lh
ill
2) T Y
sh
1
i = 1 ... m. j = 1 .... n, which we shall write as follows
s
m in (2 ). ~,
An application of combinatorics of Young tableaux
729
jsn-h 1X s respectively [ill .... i 1hl
I jl1
•••• jlm-h 1Y .... [i s1 ... iShs I jsl ... jsm-hsl Y l
TX (resp. Ty) are called standard if both the tableaux HX and KX (resp. Hy , and Ky), where
h)
...... j ...
sn- s j 1n-h
h)
...... j
KY =
1
sm- s ... jlm-h 1
are standard. Now, if TX and TV are double tableaux, we can associate to TX and Ty the "special double tableau":
TX,V
.
.
H
H'-
(KI
K) s
where s- = (sl,s2)' 51 = # boxes in H, s2 = # boxes in K. In [ 1 J it is proved that PROPOSITION: The special standard double tabZeaux TX,V such that the Zength of the longest row in H is. tess or equaZ to kl and the length of the longest row in H' is less or equal to k2 give a basis in B(k ,k }.
l
2
In order to get Cohen-Macaulayness of W(k l ,k } when k +k ~ min (n,m), in l 2 2 [ 11 it is proved that the coordinate ring of W(k ,k ) can be interpreted as an l 2 Hodge Algebra, i.e. "an algebra with straightening law" in the sense introduced in [ 31.
r,loreover, to prove normality, one can use Serre
I
5
criteri on [4 J for norma li-
ty which gives the requested result in all cases except when n=m=k l +k • In this 2 last case one gets through using Hironaka's Lemma for normality [51. One interesting aspect of the variety W(k ,k 2} is observed when k +k = min(n,m). As a l l 2 matter of fact, if this is the situation, W(k l ,k 2} can be easily seen to be the conormal bundle of the determinantal variety of nxm matrices of rank
~
kl , meaning
730
E. Strickland
by this the Zariski closure in the cotangent bundle to the space of matrices of the conormal bundle to the smooth part of the determinantal variety.
BIBLIOGRAPHY 1. 2. 3. 4. 5.
E. Strickland, On the conormal bundle of the determinantal variety, Journal of Algebra (to appear). C. De Concini, D. Eisenbud and C. Procesi, Young Diagrams and determinantal varieties, Inventiones Math., 56 (1980), 129-165. C. De Concini, D. Eisenbud and C. Procesi, Hodge Algebras, Preprint. J. Dieudonne and A. Grothendieck, Elements de geometrie algebrique, IV Publ. Math. IHES 24 (1965). M. Nagata, LocaZ Rings, Interscience Tracts in Pure and Applied Math., 13 J. Wiley, New York (1962).
Istituto Matematico "G. Castelnuovo" Citta Universitaria 00185 Roma Italy
Annals of Discrete Mathematics 18 (1983) 731-744 North-Holland Publishing Company
731
TWO CHARACTERS K-SETS WITH RESPECT TO A SINGULAR SPACE IN PG(r,q) Maria Tallini Scafati
1. INTRODUCTION Let K be a k-set in a Galois space PG(r.q) and
any given family of lines
~
in PG(r,q). The number of lines in PG(r,q) belonging to points will be denoted by t~, s = O.l, ... ,q
+
~
and meeting K in s
1, and referred to as the index s
character with respect to~.
ters.
For short. the integers t~ will be called ~-characs On the other hand. usual characters of K (i.e. the ones with respect to the
family of all lines in PG(r,q)) will be called absolute characters. PG(r.q) can be studied with respect to lines.
~-characters. ~
k-sets in
being a given family of
In order to do this. the definitions of class and type of a k-set with
respect to
~
are given as in the absolute case.
From this standpoint. a deeper knowledge of k-sets can be achieved. In this paper k-sets will be studied having two
~-characters
when
is the
~
family of all lines not belonging to a given subspace lP d (d ;;. 1); IP d will be called the singular space of K.
For short, K will be said to have two characters
with respect to the singular space IP . Furthermore. k-sets having two IR-characters d when ~ is the family of all lines in PG(r.q) not passing through a given point P will be considered.
The set of all lines in PG(r,q) through P will be called the
singular star and K will be said to have two characters with respect to the
singular star.
2. ON K-SETS IN A PLANE Let K be a k-set in
IT
q
IT
g
AND RELATED QUESTIONS
(q = ph, p a prime).
Fix any point P in
11
q
;
K will
be said to have £ characters with respect to P and to be of type (m l .m , ... ,m£)p' 2
o ~ ml < m2
1, if any line through P meets K in either ffi l • or m , ...• or ffi£ points and ml-secants, m2-secants •...• m£-secants through P do 2 ... < m£
~
q
+
732
M. Tallini Sea/ati
ex; st. When this occurs, P wiZl be eall.ed a point of indexes ml , m , ... , m£. 2 similar way the class of K with respect to P is defined.
In a
Assume P has just one character of index m, i.e. all the lines through Pare m-secants.
Then
(2.1 )
P 9" K
=>
k = m( q + 1),
(2.2)
PE K
=>
k = (m - l)q + m.
It follows: I. If two points P1 and P2 exist in ITq' both of them either
externa~
and having just one eharaeter, then the irulex with respeet to P is l ,:ndex with l'espeet to P2'
to K equa~
01'
in K
to the
Assume Pl E K and P2 f/. K and both of them having just one character, the indexes being m and n respectively. By (2.1) and (2.2) it is q(n - m + 1) = n a contradiction. I I.
,"0
~
Therefore,
two points exist in IT , one of them in K, the other one not in K, both of q
them having just one eharaoter.
Thus, if two points exist in
eharaeter, then either they both
be~ong
to
K, or they both
II
q
having just one
do not, arul their
irulexes must be equal.
Next, assume the pOint P in IT
q
is of class [m,n] , 0 p
~
m< n
~
q + 1.
De-
note by u and u the numbers of m-secants and n-secants through P 9" K and by v m n . m and vn the number of m-secants and n-secants through P E K. Then the following hold:
(2.3)
I
un
k - m{g + 1 ) n - m
I um
{g + l}n - k n - m
~
l
(
(2.4) ~J
V
n
Il vm
It
U
n
U
m
+-qn- m -~
n- m
follows:
III. If two points P and P exist having two oharaoters of indexes n, m, with
P 1
E
l 2 K. P 9" K, then n - m must divide q. 2
Assume a point P exists such that exactly one line through it is an £-se-
733
k-sets with respect to a singular space in PG(r,q)
cant, any other line through it being either an m-secant or an n-secant.
If
P fi K, wand w being the numbers of m-secants and n-secants through P, then
m
n
(
w =
k - t - mg
n
(2.5) )J
n- m
qn - k + t I wm = n - m
l
If P E K, w'
m
and w'
n
being the numbers of m-secants and n-secants through P,
then
w' m = wm --q-
(
n - m'
(2.6) )J
I w' n = Wn +-.L n- m
l
Assume three points P , P2, P exist such that P ~ K and is of class l l 3 [m,n] P (0.;; m< n,;;; q + 1), P E K and is of class [m,t,n] P (O,;;;m A SIN:> U-AR STAR IN PG (r.g) Let K be a type (O,n) k-set in PG(r,q) having
L
q
as a singular star, i.e.
any line in PG(r,q) not through P meets K either in 0 or in n pOints and such lines do exist. W.1.o.g. P may be assumed to belong to K; indeed, once such subsets are characterized, also the ones not containing P are, and conversely (it is enough to delete or to add
Pl.
739
k-sets with respect to a singular space in PG(r,qJ
When n
=q
n
+ 1
= 1, K obviously consists of P and another point Q r P. Since
is impossible, the assumption
(5.1)
is made on what follows.
4. Consider a prime TI not through P such that TI n K r~. The k'-set K n TI is of absolute type (D,n) (as n < q + 1). Since the dimension of TI Assume r
r - 1 lP
~
~
3, by [4) prop. IX, n = q and K n TI = TI \ lP
r-
2'
Any prime TI' through
r- 2 , but not through P, either is external to K, or is such that K n TI'
TI' \ lP r _2· If a prime TIo ( B P) through lP r _2 exist which is external to K, then (as n = q) K = P.l (r,q) \ TI. Except this case, any prime TI through lP 2' o rother than the prime joining P with lP r- 2' belongs to K. It follows that =
I
K = P.l (r,q) \ lP
r-
l'
Thus, the following result has been proved.
XV. Let K be a k-set of type (D,n) in P.l(r,q), r through P as a singuZar star (n
~
2, P E K).
~
4. having the set of aZl lines
Then n = q and K = P.l(r,q) \ TI.
where TI is a given prime.
Now, assume the dimension is r = 3. a point Q of K, Q r P, and t
Let r be a line through P containing
= \K n r\. Consider the points in K on the lines
through Q (any of these lines other than r being an n-secant), then k
(5.2)
=
2
(q + q)(n - 1) + t .
Let s be an n-secant of K not through P.
Since n
~
q, any plane through s,
but not through P, meets K in a set of absolute type (D,n); therefore, \K n TI\
= (q
+ l)(n - 1) + 1.
Let
a
be the plane jOining s with P.
Consider the
points of K on the planes through s, then k
\a
Hence, setting c (5.3)
= ((q
C
+
l)(n - 1)
+
1 - n)q
+
\a
n K\.
n K\,
\a
n K\
= k - q2(n - 1) = t
+
q(n - 1).
Equality (5.2) shows also that a line through P meets K either just in P or in
!
= k - (q2
q)(n - 1) points. Let r be an t-secant of K through P; any plane through it either contains no +
other point in K, or it contains another point of K; in the latter case it contains an n-secant not through P; thus - by the previous argument - it contains
740
M. Tallini Scafati
exactly c points of K, where c is given by (5.3). planes of K through r. k
= a(c -
lows.
Let a be the number of c-secant
Then (considering the points of K on the planes through r)
t) + £, from which (taking into account (5.2) and (5.3»
Therefore, all planes through rare c-secant.
a
=q
+ 1
fol-
Thus, the following is
proved: any plane through P either meets K exactly in P, or is a c-secant plane of
K and so it contains exactly being an t-secant of K.
* k -set
v
= (c - 1)/(£ - 1) lines through P each of them
In the projective plane E , consisting of the lines through P, consider the I< P ~ K formed by the t-secants of K. By the previous argument, K' is of type 1,
either (v) or (O,v).
In the fanner case, K is coincident with l: , i.e. any 1ine 2 p through P as an t-secant of K; then (q + q + 1)(£ - 1) = k - 1, so that £ = n, by hence, K is of absolute type (O,n).
(5.2);
the complement of a plane not through P. v
Thus, by
prop. IX, n = q and K is
[4)
In the latter case, (Q > 0),
= (c - 1)/(£ - 1) = pQ
from which, taking into account (5.3), c - 1 = (t - l)pQ = £ - 1
+
q(n - 1) ,
so that
(5.4)
(£ -
n -
l)(pQ - 1)
(£ -
q
l)(v - 1) ,q
Since q and pQ -
are coprime, q must divide
integer; but £ -
~
£
1, i.e.
£ -
1 = bq, b a positive
q; therefore, n = v and K is a cone projecting from P a set
in the plane whose absolute type is (O,n).
Thus, the following proposition is
proved. X VI. In P.i ( 3, q) a k-set K of type (0, n) having a singulCU' stCU' whose centre is P (n > 2, P E K) is either the compZement of a plane not through P or a cone
projecting from P a set of absoZute type (O,n)on a plane not through P.
6. ON TYPE (m,n) K-SETS WITH SINi ULAR PRN E IN P.i (r ,q) Let K be a type (m,n) k-set with singular prime any line not belonging to exist.
II
II
in ffi(r,q), r;;' 3, i.e.
meets K either in m or in n points and such lines do
Since in sect. 4 k-sets of types (O,n) and (m,q + 1) (taking their
complements) with singular prime have been studied, in what follows
k-sets with respect to a singular space in PG(r,q)
741
l..;;m 1 (by
Let a be the plane joining Q with r.
meets K in a k'-set of type (m,n) with r as singular line.
case, a plane
U IT)
By prop.
Thus, in any
through r exists meeting K in a k'-set of type (m,n) with r as
a
singular line. Therefore, taking into account sect. 3, the following hold (see (3.6). (3.10)):
n
(6.2) r
(6. 3) ~
l
n
=
=£
m + pa,
+ (s + l)p
1.;;
a .;; h -
1
(a an integer),
a
a h-a m = £ + sp ,- (p + 1)
ix; 4) every conic of V is not incident with d; 5) V contains
q2 conics;
projectivity
3 of r, then there exists at least one 2 such that VP = V' .
if V ~ld V' are two V
6) p
of
L
3 satisfies the following conditions: a) d is a line of 2 b) d' = an E' is a line of Sand d' n C =~. We can prove: Assume that V=V
$;
PROPOSITION 4 ([20], teor. 1.1, prop. 1.3): The variety V is the intersection ,apQl't [r'om a tangent plane) of the quadric cone with vertex d and having the
coni c C as directrix and of the quadric cone with vertex a point 0 ;,,"ving
,!
regulus of S as directrix.
E
l: - E'
and
777
Varieties of PG( 4,q), subplanes of translation planes
THEOREM 1 ([20], teor. 2.1, 2.2; coroll. 2.2. Nota 2.1): The var>ieties V of satisfying a) and b) r>epr>esent pr>eaisely the non-affine Baer> subplanes of
J:
TI.
Using the projective properties of such varieties v 3 of J: and starting from 2 them, we are able to construct spreads of hyperplanes of E and we can prove that any such a spread is regular.
Therefore we can state:
THEOREM 2 ([2], teor. 2.4): Any var>iety V 3 of E r>epr>esents a non-affine Baer> 2 subplane of a tr>anslation plane
TI
if and only if
IT
is desarguesian.
At this pOint, to find examples in the non-desarguesian case we can try to answer to the following question: since such varieties of E can be constructed using a projectivity
~
: d
4
C, what does it happen if we don't require that
~
is
a birationa1 isomorphism? Let F=GF(q).
= PG(4,q) , q=s2 and let cr : x
J:
We construct a variety V of
J:
4
S X
the involutorial automorphism of
with the following properties:
a) V contains a line d; b) V contains an algebraic curve C of order s+l. c) if a is the plane of C, then and =
~;
d) there exist two points, P and Q, on d
such that V contains two lines r=PP' and r'=QQ' such that P'fQ' and P' ,Q'
E
C; e)
V is the set of the points incident with the following set G = {r,r', v~(a) /
V
a
E
C- {P',Q'}} where each v~(a) is a curve of order sand
incident with P. The geometric properties of such a variety V are summarized in the following proposition: PROPOSITION 5 ([ 21] , prop. 2.2; teor. 2.1): ]) the lines r and r' are mutually s s skew; 2) for> every a E C- {P',Q'} we have: r n Vl(a)=P and r' n V1(a)=~; 3) eaah var>iety V~(a) interseats C in one point; 4) the number> of the points of V is
(q+1)2. Assume that S is a spread of a hyperplane E' of E which represent a translation plane
TI
over a Knuth's semifie1d S, where the field F is simultaneously the
left, the middle and the right nucleus of S (see [15] pp. 215-216). Moreover, assume that d is a line of S, d' =a n E' is a line of Sand d' n C = 0. Then we can prove:
R, Vincenti
778
PROPOSITION 6 ([ 211, prop. 1.5, 1.6, 1.7, 2.1): The variety V is the intersection (;::tpar1; ·:lUI'V':,
';I'I.;i
[1'01'1
a r:ang6n::
p~ane)
of the cone of order s+l 1.Jith vertex d and having the
C Cl d, the last condition holds for all j when it holds for j=d+1). In [8] B. Segre stated the following problem: PROBLEM 1: For given d and q, what is the maximum value of n (not. s(q,d))
for
which n-arcs exist in PG(d,q)? Partial solutions given by B. Segre [8], J.A. Thas [9,10,11], L.R.A. Casse [1], H.R. Halder and W. Heise [3] are stated in the following table:
790
M.L.H. Willems
q
d
s(q,d) q+l
(trivial)
even
2
q+2
[ 20]
odd
2
q+l
[ 8]
q>3
3
q+l
[ 1 ,8]
q>4
4
q+l
[1,8]
q > 4 + h, q even
4+h, h > 0
~+h+l
[ 1]
q> 5
q-4
q+l
[ 11 ,3]
q > 3, q odd
q-3
q+l
[ 11 ,3]
Q > 3, q even
q-2
q+2
[ 9]
Q> 6
q-5
q+l
[ 3]
q-l
q+l
[ 3]
q+h, h ;;. 0
q+h+2
[ 3]
n
q+l
[ 10]
q-n-l
q+l
[ 11 ,3]
Q odd, Q> (4n-5)
q odd,
q
> (4n-5)
2 2
2. OPTIMAL CODES Let F denote the aZphabet {A l , ••. 'An} of n symboZs, n ;;. 2. The set V=Fm, m;;. 2, is called the set of words. If j E {l, ... ,m} and Pj: V -+ F: (xl,,,,,xrrrxj' then p. is called the j-th coordinate function of V. J
The
H~ling
A code C is a subset of V.
distance between 2 words u,v of C is the number of places where they
differ and is denoted by d(u,v).
The minimum distance d of a code C is the
minimum Hamming distance between its code words. Assume that i ,.;; m, i
E
N.
The set A = {jl ,j2"" ,ji } is called
information digits of the code C iff {(po (c), ... ,p. (c)) : c J1
J.
E
~
set of
C} = Fl.
A code
C, C c V, with cardinality ni, is called a (m,i)-cod~ of order n iff C has at least one set of i information digits.
A (m,i)-code of order n is called an
optimal code iff every set of i indices is a set of i information digits.
Thus,
an optimal (m,i)-code of order n is a code with word length m for which every i symbols may be taken as message symbols.
Clearly, an optimal (m,i)-code of order
n has minimum distance d=m-(i-l) between its code words. Assume now that q is a power of a prime.
A (m,i)-code Cover GF(q) is
called ':inew' iff C is a subspace of the m-dimensional vector space V over GF(q). A linear (m,i)-code over GF(q) with minimum distance d is called an
[m,i,~
-code
Restricted Mi-, Li-spaces, optimal codes and n-arcs
over GF(q).
791
R.C. Singleton [16] proved: If C is an (m,i)-code over GF(q) with
minimum distance d, then d ~ m-10g q qi+1• The (m,i)-codes over GF(q) with minimum distance m-i+l are called maximal distanee separable eodes (MDS eodes). Thus, if C is an optimal (m,i)-code over GF(q) then C is a maximal distance separable code.
Further, F.J. Mac Williams and N.J.A. Sloane proved the following
theorem in [16]. An [m,i,d]-code Cover GF(q) is a MDS code iff C is an optimal (m,i)-code over GF(q). One can state here the following problem: PROBLEM 2: Given i and q, find the largest value of m for which an [m,i,m-i+1]code exists over GF(q).
There is an extensive literature on this problem but we
restrict ourselves in stating the following results (resp. [16] and [16]): For i
= 3 and q odd, we have m ~ q+1. Moreover, there exists an [q+1,i,q-i+2]
cyclic MDS code over GF(q). For i
= 3 and q even, we have m ~ q+2. Moreover, there exists an [2 h+2,3,2 h]
Reed-Solomon code. For other values of i see F.J. Mac Williams and N.J.A. Sloane [16] . W. Heise [4] noticed the equivalence of B. he proved the following theorem: Let m ~ i a prime.
~
Segre's problem 1 and problem 2;
2 be integers and Zet q be a power of
Then there is an [m,i,m-i+l]-eode over GF(q) iff there is an m-are in
PG(i-l,q) (thus, iff m ~ s(q,i)).
The relation between the code and the m-arc can be described as follows [16]: Let G.l,m be the matrix whose columns are coordinate vectors of the m points of an m-arc in PG(i-1,q). Then the code C (over GF(q)) which is generated by the rows of the matrix G.l,m is an [m,i,m-i+l]-code over GF(q). By 1 we have: (a) There is at Zeast one [q+3,4,q]-eode over GF(q) iff q=2.
(b) There is at least one [q+2,4,q-l]-eode over GF(q) iff q
E
{2,3}.
3. RESTRICTED i-SPACES We define a linear spaee (LS) as a finite set of v elements, called points, of which b subsets, called lines are distinguished so that the following hold: (LS1) Any 2 distinct points u,v belong to exactly one line uv.
792
M.L.H. Willems
(LS2) Every line contains at least 2 points. (LS3) b > 2. N.G. De Bruijn and P. Erdos [2] proved the following celebrated theorem: If L is a linear space with v points and b lines, then b > v. A line containing k points is called a k-line. 2
number n defined by n .;;; v < (n+l)2.
The order of a LS is the
The LS is called restri-cted iff it satisfies:
(RLS) (b_v)2 ~ v. Now we define some important LS's.
Notations and terminology are taken from
A near-pencil is a LS with all its points but one collinear.
[15]
is the unique RLS with 6 points having one 4-line and one 3-line.
Lin's cross
A finite semi-
affir.>3 l,l.ane of type III (FSP3) is the LS obtained from a finite affine plane
(FAP) by adjoining to it one "infinite" point.
If the FAP we started with, has
order at least 3, and if we delete a "finite" paint from this FSP3, then we obtain a d6 ;';i:tte plane (WP).
A LS is called an inflated FAP if and only if the follow-
ing conditions hold: (a) a subset V of its point set together with the set of induced lines, form a FAP, say L'\
(b) the complement of V together with the set of induced lines, form a LS, say L'; (c) any line of L joining two points of L' is completely contained in L' ; (d)
~':
any 1i ne of L joining a point of L' and a point of L contains at least 1:
one more point of L
It is readily seen that the number of points of L' is at most the number of parallel classes of L1'.
If L' is a near-pencil, then L is called a simply
inflated FAP; if L' is a finite projective plane (FPP). then L is called a projectiveZy inflated FAP.
The following result is due to J. Totten [15]: L is a RLS if and only if L is one of the following: (i) a near-penCil; (ii) Lin's cross; (iii) a FAP or a WP; (iv) a FSP3, or a FPP of order n with at most s
~
n points deleted, which are
not collinear if s : n; (v) a simply inflated FAP, or a projectively inflated FAP. We note that in our context a restricted linear space will also be called a
Restricted Mi-,
Li~paces,
optimal codes and n-arcs
793
restricted DO-space.
Suppose that K = (P,B,I) is an incidence structure with P and B disjoint sets and I
C
P
B an incidence relation.
x
Assume that B = Bl
B3 , with Bl ,B 2 ,B 3 mutually disjoint sets. Suppose that B3 i ~, but possibly Bl = ~ or (and) B2 =~. The elements of Bl U B2 (resp. B3) are called lines (resp. circles). Two points of K are called collinear iff they U
B2
The elements of P are called points.
U
are incident with at least one line of K. If Pl, ••• ,Pi are i pairwise non-collinear points of K, then the incidence structure
P \ {x
P : x is collinear with Pj , j
E
B = (B ) Pl···Pi ' P,,,.Pi (Bl)Pl",P
Pl",Pi
U
E
{l,,,.,i})}
{L E
B2
E
{l, ... ,i})}
B3
E
= I n (P
'(LIPj,j
{l,,,.,i}},
(B ) , 3 P''''P i
Bl : '(LIPj,j
(B 3)P ",Pi = {C l I
(B ) 2 P,."Pi
{L E
i
(B 2)P ",P l i
U
E
Pl",P i
x
6 ) is called the i-RS (i-residuaZ space) of Pl " 'P i
A restricted Di-space (RDi) (resp. restricted Laguerre i-space (RLi), restricted Minkowski i-spaae (RMi)), i J = (P,B=B
1, is an incidence structure
B3 , I) with P,B l ,B 2 and B3 mutually disjoint sets, B3 f~, Bl U B2 = ~ (resp. Bl f ~ and B2 = ~, B, f ~ and B2 f ~), P the point set, B, U B2 the line set, B3 the circle set, and I C P x B the incidence relation, satisfying l
U
B2
~
U
the following: (i) if B. f 1
~,
i
E
{1,2}, then each point of P is incident with exactly one
1i ne of B ; i
(ii) each two lines of different families (i.e. two lines of resp. B, and 62) are incident with at most one common point; (iii) each circle of B3 and each line of B, common point;
U
B2 are incident with at most one
794
M.L.H. Willems
(iv) each i+2 pairwise non-collinear points are incident with exactly one circle of J; (v) each i-RS of J is isomorphic to a restricted linear space; (vi) each circle (resp. line) of J is incident with at least i+2 (resp. 2) points of J. Mostly we shall identify a line (resp. circle) with the set of all points incident with it.
It is clear that the Mobius i-structures (resp. Laguerre i-
structures, Minkowski i-structures) of order n [5J, i
~
1, are the restricted Di-
spaces (resp. Li-spaces, Mi-spaces) for which every i-RS is an affine plane of order n. We defi ne a "special" Laguerf'e i-structure [19J of order n, i
~
1, as a
restricted Li-space for which every i-residual space is a projective plane of order n, minus 1 paint.
One can easily prove that the order n of a "special"
Laguerre i-structure is even [ 19) • In [14J we proved the following classification theorem: The incidence structut'e'
0'
~s
a restricted Li-space, i
~
1, iff J is one of the foHowing:
\' 1 J a wguerre i-structure of order n;
(2) the i'1eide'1ee structure obtained from a "special" Laguerre i-structure f:
L oj Drdel' n. n even, by delet'ing s points, s .;;; (n-l)(n+i+J)/(n+]), for which: (1~)
at ,','IOSt '1-2 (l'esp. '1-1) of the deleted points are incident with a common line
* (ii) every '1+1 Z-ines of L* contain at most n-1 of the s (l' 1, and n
~
2, of order n, then
n is even.
In the first part of section 4 we noticed that there exists a classical "special" Laguerre i-structure of order q iff there exists a [q+;+l, ;+2, q]-code over GF{q).
More precisely, here we prove
THEOREM 4: The "speciaZ" Laguerre i-strueture L{C) of order q whieh eorresponds to the optimal (q+i+l,i+2)-eode Cover GF(q) is elassieal if C is linear.
Con-
versely, if the "speeial" Laguerre i-strueture L of order q is eZassieaZ, then at least one of the eorresponding optimal (q+i+1,i+2)-aodes C{L) over GF{q) is linear.
Restricted Mi-, Li-1Ipaces, optimal codes and n-ilrcs
799
PROOF: Assume that the optimal (q+i+1.i+2)-code Cover GF(q) is linear.
By sec-
tion 1 there exists a (q+i+1)-arc 0={P1' .... Pq+1+ . 1} in PG(i+1,q). such that the (i+2,q+i+l)-matrix M=(X ••. X . 1)' where the column X. is coordinate vector of 1 q+1+ J p. , j En, ... ,q+i+1 }, in PG(i+1.q). is a generator matrix of the code C. Assume J
that PG(i+1,q) is the hyperplane x.1+ 3=0 in PG(i+2,q). Let t denote the point t(O .•. Ol) of PG(i+2,q), and let to denote the cone in PG(i+2,q) with vertex t and generators tPl •••.• tp q+1+ . 1.
Let c denote an arbitrary word of C.
Then we define
the following bijection f of P (the point set of L(C) onto the point set of the cone to: Vj
E
{l, •••• q+i+l}, VA
E
GF (q) : f(A.j) = t(O •.• Ol)(A-p.(c))+t(x. 0). J
J
This implies that a line GF(q)x{j} of L(C) is mapped onto the set tp.\{t}. Con. 2 J sider now one of the q1+ hyperplanes a of PG(i+2,q) which are not incident with i+2 t. Then a is given by the equation '~l a.x.+x.+ =0. The intersection point q. of t· i+2 J- J J t 1 3 J a with tp. is given by (X. 0) -kL1 ak(X')k (0 ••• 01), with (X')k the k-th coordiJ J = J i +2 J nate element of Xj • This implies that f-l(qj)=(Pj(c) + k~l ak(Xj)k • j). Since C is a linear code whose words are all possible linear combinations of the rows of M and c E C, we have that f
-1
(a n to) is a circle of L(C).
This implies that every
circle of L(C) is mapped by f onto a hyperplane section of to in PG(i+2.q). L(C) is isomorphic to L(O).
Hence
This implies that L(C) is classical.
Assume now that the "special" Laguerre i-structure L of order q is classical (i.e. is isomorphic to L(O). with 0={P1' ••• Pq+1+ . 1} an
(q+i+1)-arc in PG(i+l,q)).
The vertex of the corresponding cone in PG(i+2,q) is denoted by t. as follows that one of the codes C=C(L(O)) is 1inear.
One can check
Assume that PG(i+2.q) is
coordinatized as follows: t=t(O ••• Ol). PG(i+1.q) : xi +3=0 , Plt(X10)' •••• Pq+i+1t(Xq+i+l 0). Let Kj denote the line tp. , j En •...• q+i+l}, and let x(k,j) , k E GF(q) denote the point of K. \ {t} J
with coordinate vector 1
cl=kl.··k
1
t
2
t
(0 ••• 0 l)k+ (X. 0). 2
J
Assume that
J
. land c2=k l ••• k . 1 are 2 code words of C. such that c 1 (resp. c2 ) q+1+ q+1+ i+2 1 corresponds to the section of to and the hyperplane Hl : '~1 a.x.+x. 3=0 (resp. i+2 2 J- J J 1+ H2 : .L a.x .+x. 3=0) in PG(i+2.q). Obviously, the hyperplane H has also the J= l J J 1+ R, equation:
M.L.H. Willems
800
Xl
xi+2
xi +3
(X ) I\ j 111 )
(X. )
k~
[
O.
Jl
1\ J 11) i+2
)
lX ji +2J i+2 with
{l,2} and {jl, ... ,ji+21
i E
C
(l, .... q+i+l}.
first i+2 indices are information digits. one word c for which p.(C)=Ak~+~k~ • j denote p (c). 5
J
J
J
Since C(L(O)) is optimal the
So, for any
E {l, ... i+2}.
We shall prove now that y =Ak 5
1
2
S
s
+~k
E GF(q) C has exactly
A.~
Let
• s E {i+3, ... ,q+i+l} s It is clear that the hypery
i+2 1 2 which corresponds to c is given by .L a.x.+x. 3=0 with a.=Aa.+~a. J= l J J 1+ J J J j E (l, ... , i+2 ). Let us determine now the intersection point x (y ,s) of a with s s i+2 1 2 1 2 .L (Aa.+~a.) (X ).+y=O. So y =Ak +~k This implies Ks' S E {i+3, ... ,q+i+ll s s s J= l J J s J s that C(L(O)) is linear, and so the theorem is proved. plane
a
BIBLIOGRAPHY 1. 2. 3. 4. 5. 6. 7. 8.
L.R.A. Casse, A solution to B. Segre's problem I ,VII.Osterreichischer Matizematikerkongress, Linz, 16. bis 20, SePt~mber 1968. N.G. De Bruijnand P. Erdos. On a combinatorial problem, Indag. Math., (1948) 10. 421-423. H.R. Halder and W. Heise, On the existence of finite chain-m-structures and k-arcs in finite projective spaces. Geometriae Dedicata, 3, 483-486, 1975. W. Heise. Optimal Codes, n-Arcs and Laguerre Geometry, Acta Informatica 6, 403-406, 1976. W. Heise and H. Karzel, Laguerre- und Minkowski-m-Strukturen, Rend. Ist. di Matem. Univ. di Trieste, Vol. IV, fasc. II. 1972. W. Heise and H. Seybold, Das Existenzproblem der Mobius-, Laguerre- und Minkowski-Erweiterungen endlicher affiner Ebenen, Verlag der Bayerischen Akademie der Wissenschaften, MUnchen, 1975. W. Heise, Teoremi di non existenza di codici ottimali e m-strutture di Laguerre, Atti Sem. Mat. Fis. Univ. Modena, XXVII, 1978, 45-50. B. Segre, Curve razionali normali e k-archi negli spazi finiti, Ann. Mat. Pura Appl. 39. 1955, 357.
801
Restricted Mi-, Li-lJpaces, optimal codes and n-arcs
12.
J.A. Thas, Normal rational curves and (q+2)-arcs in a Galois space Sq-2,q (q=2 h), Rend. Accad. Naz. Lincei (8) 47 (1969), 249-252. J.A. Thas,.Normal rational curves and k-arcs in Galois spaces, Rend. di Mat. (6) 1, (1968) 331-334. J.A. Thas, Connection between the Grasmannian Gk- l ' n and the set of the k-arcs of the Gal oi s space Sn_q' Rend. di Mat. (6) 2 (1969) 121-134. J.A. Thas and M.L.H. Willems, Restricted Di-spaces, Part I, 29 pp. (to
13.
J.A. Thas and M.L.H. Willems, Restricted Di-spaces, Part III, 25 pp. (to
14.
J.A. Thas and M.L.H. Willems, Restricted Li-spaces, Part II, 30 pp.
9.
10. 11.
appear). appear). (to
appear).
18.
J. Totten, Classification of restricted linear spaces, Can. J. Math' J 28 (1976), 321-33. F.J. Mac Williams and N.J.A. Sloane, The theory of error-correcting codes North-Holland Mathematical Library, 1978. M.L.H. Willems, Beperkte D-J L- en M-ruimten J Ph. D. Thesis, State University of Ghent 1980. M.L.H. Willems and J.A. Thas, Restricted Di-spaces, Part II, 22 pp. (to
19.
M.L.H. Willems and J.A. Thas, Restricted Li-spaces, Part I, 30 pp.
15. 16. 17.
J
appear). appear).
20.
B. Segre, Lectures on modern geometry, Cremonese, Roma (1961).
Dienst Meetkunde R.U.C.A. University of Antwerp Groenenborgerlaan 171 2020 - Antwerp Belgium
(to
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Annals of Discrete Mathematics 18 (1983) 803-804 North-Holland Publishing Company
803
.-
SUR DES CLASSES DE GROUPES FINIS RESOLUBLES Guido Zappa
Rappellons qu'une classe F de groupes finis resolubles est appelee olasse de Fitting si
a) (G
E
F, N sous-groupe normal de G)
~
NE F;
b) {G=N 1N2 , Ni E F, Ni sous-groupe normal de G (i=l,2)) * G E F. Tous les groupes que nous considerons sont supposes finis et resolubles. Si Fest une classe de Fitting de groupes, un sous-groupe H d'un groupe G est dit F-maximal si H est un element maximal dans 1 'ensemble des sous-groupes de G qui appartiennent a F. L'union des sous-groupes normaux de G qui appartiennent groupe G qui est aussi normal dans G et appartient
a F:
a F est
un sous-
l'F-padioal de G.
Une classe de Fitting est appelee nOPmale si pour tout groupe G, l'F-radical G est F-maximal. Les classes de Fitting normales ont ete etudiees par beau coup d'auteurs. l-1ais la definition de classe de Fitting normale est d'une
fa~on
dHferente de celle
de classe de Fitting (ou de celle de formation). En effet, la definition de classe de Fitting a un caractere constructif, c'est-a-dire, elle nous dit que si des groupes donnes appartiennent de Fitting F, aussi d'autres groupes que nous pouvons construire sont dans F. constructive.
a une classe
a partir d'eux
La definition de classe de Fitting normale, au contraire, n'est pas 11 est interessant de donner une definition constructive aussi pour
les classes de Fitting normales. Nous dirons qu'une classe de groupes F vepifie la propriete
~)
si pour tout
group G verifiant les conditions suivantes a), b), c): a) G=HM avec H sous-groupe F-maximal de G, Msous-groupe normal maximal de G, H n 1·1 normal dans G; b)
H ne centralise pas M l·lnH MnH
804
G. Zappa
c) pour tout sous-groupe normal N de G tel que '·1 () HeN ,
_H_ centralise 1-1ri H
N , t~
~1,
() H
est verifiee aussi la condition: d) ME F
Nous avons demontre le suivant TH€ORb~E: Vne cZasse de Fitting est nO!'l7laZe si et seuZement si elle vel'ifie la ?!'opridte a).
La propriete a) a un caractere constructif. De ce theoreme on retrouve facilement le corollaire suivant: (Cossey) 7'oute classe de Fitting nonnale non tl'iviale contient Za eZasse des groupes finis nilpotents.
Le theoreme precedent conduit
a introduire
le concept de classe normale.
Une classe de groupes Fest appelee classe normaZe si: a.) (GE F,
rl
b) (A, BE F)
sous-groupe normal de F) ~
A
x
~
(NE F);
BE F;
c) F verifie la propriete a). On demontre facilement que: Toute cZasse normale contient to us les groupes abeZiens elementaires.
PROBLb~ES: 1) Voir si 1 'intersection de classes normales est une classe normale.
2) Determiner les classes normales minimales non triviales. Les demonstrations sont dans rna note: "Un'osservazione sulle classi di Fitting normali" qui va paraitre dans les "Rendiconti dell'Accademia Nazionale dei Lincei".
Istituto t·1atematico "U. Dini" Viale tlorgagni, 67A 50134 Firenze Italy
Annals of Discrete Mathematics 18 (1983) 805-818 North-Holland Publishing Company
805
FINITE NON-EUCLIDEAN PLANES H. Zeitler
Starting with finite Miquelian Mobius-planes of odd order. we construct "Poincare-models" of finite hyperbolic planes. For these planes some theorems are given. They include counting-statements. the three-reflection-theorem and the theorem about the minimal decompositionlength of reflection products.
Finally we show a connection between these hyper-
bolic planes and the so called BL-planes. This lecture essentially is a survey of results.
Missing proofs can be
found in the cited papers by the author.
1. BL-PLANE (KLEIN-MODEL) At first I'll give a short account of a paper by T.G. Ostrom, written in 1962.
1.1 AN OVAL Let 0 be an oval in projective plane PG(2,q) over a finite field K = GF(q). According to B. Segre the ovalO is even a conic. bei ng a prime number and e
E
As you know q equals pe with p
:IN.
Let us restrict ourselves to the case where q is odd, hence Char Kt 2. The ovalO contains exactly q+l points.
There are exactly (q;l) secants.
exactly q+l tangents and exactly (~) passants (exterior lines) of O.
All the
points on the tangents other than the respective touching point are called expOints; in the same way the points of the oval are called on-points and all the other points are called in-points. There are exactly (~) in-points. contains exactly t(q-l) and each passant exactly t(q+l) in-points.
Each secant Through each
in-point there are exactly t(q+l) secants and just as many passants.
H. Zeitler
806
1. 2 BL -PLANES
Let's now construct BL-planes, according to T.G. Ostrom. BL-points: in-points of O. BL-lines:
all the BL-points of each secant and of each passant of
a form
a BL-
line. let's presuppose in this paper incidence is incidence in the common sense. Using symbols that are common in combinatorics we obtain the following results for these new structures, for these BL-planes: v 1
1
q
1
=
(~);
1
1; b = (q;l); l Through any Bl-
A =
kl = Z(q-l); r l = Z(q+l); b2 = (2); k2 = Z(q+l); r 2 = Z(q+l). point that does not lie on a given BL-line there are either exactly 01 exactly 02
=
1
Z(q+3) or
= ~(q+l) lines, which do not intersect the given line. In each case
there are at least two non-intersecting lines.
To remember Bolyai and Lobatchef-
sky, T.G. Ostrom called these planes BL-planes. But we won't follow this line of geometry at this point and just make two remarks. 1. T.G. Ostrom also investigates non-desarguesian projective planes. 2. E. Seiden has also constructed finite BL-planes in the case that Char K = 2. Basically the BL-planes are the well-known Klein-model of the classical noneuclidean geometry, applied to a finite projective plane. In a review of finite hyperbolic spaces in 1971, R.J. Bumcrot asked the question: "Is there a finite analog to Poincares-half-plane-model of the classical hyperbolic planes?" Now - and this is the aim of our talk - we'll try to give an answer to this question.
2. THE (K,L)-PLANE The starting point in developing the classical Poincare-half-plane-model is the Gauss-plane over the special pair of fields (IR, (t) • 2.1 THE PAIR OF FIELDS (K,L) In complete analogy to the Gauss-plane we'll now consider the plane over the pair of fields (K,L).
Let K be the finite field GF(q) with q
= pe. Further, but
only for the purposes of this talk, let's assume that Char K is unequal 2 - as
807
Finite non-euclidean planes
already stated.
Then let the field L rise from K by a quadratic separable ex2
,~
tension using an irreducible polynom of the form x + b with b E K. we obtain L = K(£) with £2
In this way
For all elements X E L we have X = Xl + £x 2 The element X = xl - £x 2 E L is called conjugate to the element +
b = O.
with x ,x E K. l 2 X. The mapping X ~ X for all X E L is the only non-identical automorphism of L, which fixes K element by element.
The set of all automorphism of L is called Aut
These mappings may be represented by X ~ xP with pE {l,p,p2, ... ,p2e-l} for all 2 X E L. All these mappings fix the field K as a whole. The element XX = x + 1 2 + bx 2 E K is called the norm of the element X = xl + £x 2• We write N(X). It L.
follows N(XY) = N(X)N(Y) for all X,Y ELand N(l) = 1.
Further we obtain N(X)
if and only if X = O. All the elements of K are norms of elements of L. *2 1 we have IK I = 2(q-l).
=0
Finally
2.2 THE ELEMENTS OF THE (K,L)-PLANE (K,L)-points
P=LU{P",,} P is called the "unproper" point 00
(K,L)-lines
{X E L/XM + XM + d = O} U {P,,) with MEL", dE K
(K,L)-circles
{X E L/N(X-M) = c} with MEL, c E K*
The set of all (K,L)-lines together with the set of all (K,L)-circles is called the set of (K,L)-cycles, we write Z.
The structure (P,Z,E) is called
(K,L)-plane. According to W. Benz these are exactly the Mobius-planes of order q in which both the theorem of Miquel and the so called "touching-pencil-theorem" hold.
In the (K,L)-planes we have : v
= q2+1; bl = q(q+l); b2 = q2 (q-l);
2 b = b + b = q(q +1); k = q+l; r = q+l; A 11:1 q+l. l 2 cycle through three (K,L)-points.
There is exactly one (K,L)-
2.3 (K,L)-CYCLE-TRANSFORMATIONS Each transformation which maps (K,L)-points one to one onto (K,L)-points and (K,L)-cycles one to one onto (K,L)-cycles and at the same time preserves is called a (K,L)-cycle-transformation.
These transformations are represented by
the following equations. X' =
incidenc~
~~~~~:~ with X,X',S,T,U,VE L; SV - TU
* 0;
p(x) E Aut L.
In respect of the point Poo we'll give special definitions.
808
H. Zeitler
for X = Pro and U = 0 for X = Pro and U * 0 ( Pro for Up(x) + V = o and U * O.
J ~ro i U
X•
For the determinant we obtain SV - TU E L* . L*2 (it is determined up to L*2 ). M
denotes the set of all (K,L}-cycle-transformations.
'J(X)
id we speak of (K,L)-homographies and we write H.
~(X)
Xare
In the case that
The mappings with
called (K,L)-antihomographies.
The mappings represented by X' = up sP~~~+t +v with X,X' E L; s,t,u,v E K;
* O;p
sv - tu
E Aut L are exactly those (K,L)-cycle-transformations with the
= {X E L/X - X = O}
line r
U
{Pro} as a fixed line.
there are special definitions again. K;' . L: 2, BotZ. UMI, 9 (1974), 188-196. T.G. Ostrom, Ovals and finite Bo1yai-Lobatschefsky planes, Amer Math. MonthZy, 69 (1969), 899-901. B. Segre, Su11e ova1i nei piani 1ineari finiti, Rend. Aae. Naz. Linaei VIII (1954), 141-142. E. Seiden, On a method of construction of partial geometries and partial Bo1yai-Lobatschefsky planes, Amer. Math. Monthty, 73 (1966), 158-161. H. Zeitler, Uber (K,L)-Ebenen Dissertation, Kassel (1977). H. Zeitler. Hyperbo1ische Ebenen Uber Korperpaaren, AnnaZes Universitatis Saientiarum Budapestiensis XXIV (1981), 63-85.
Universitat Bayreuth Mathematisches Institut Postfach 3008 8580 Bayreuth Federal Republic of Germany
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Annals of Discrete Mathematics 18 (1983) 819-820 North-Holland Publishing Company
819
OTHER TALKS S. Abeasis : Representations of quivers and their semi-invariants. S. Antonucci
Simple and generalized colouring and multicolouring of cliques without a given number of Hamiltonian cycles and of regular graphs.
L.M. Batten
Jordan - Dedekind spaces.
M. Biliotti and G. Menichetti : Elations of derived semifield planes. L. Borzacchi ni : Convers ion rna tri ces and graph-reconstructi on. T. Brylawski
Hyperplane reconstruction of the Tutte polynomials of a matroid.
I. Debroey : Non-trivial
r~-regular
graphs.
C. De Concini : Special bases for the symmetric groups. M. Dehon : On the subdesigns of an SA(2,3,v). K.J. Dienst: Generalized quadrangles in projective spaces. L. Dubikajtis : Relation of semi-betweennes. J. Ch. Fischer and J. Shilleto : Perpendicular polygons. T. Grundhofer : Groups of projectivities of egglike Mobius planes. M. Limbos: A connection between some rational curves and affine spaces. G. Lunardon : Insiemi indicatori proiettivi e fibrazioni planari di uno spazio proiettivo finito. H. LUneburg : Generalized Andre planes which are almost Andre planes. M. C. Marino and L. Puccio: On parameters of Ls-colorations of nondirected and finite graphs G. Menichetti : Finite commutative (non associative) division algebras. P. Plaumann and K. Strambach : Partitions of Lie groups. C. Procesi : Springer representation and symmetric functions. R. Rink: A class of totally disconnected topological projective nearfield planes. I.G. Rosenberg: Regular and strongly regular selfcomplementary graphs. C. Somma: Generalized quadrangles with parallelism. L. Teirlinck : Embedding properties of matroids and linear spaces into projective spaces. P. Vanden Cruyce : Convexity in graphs.
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821
Annals of Discrete Mathematics 18 (1983) 821-824 North-Holland Publishing Company PARTICIPANTS L.M. Abatangelo
Bari
G. Cohen
Pari s
V. Abatangelo
Bari
C.J. Col bourn
Saskatchewan
S. Abeasis
Roma
M.J. Col bourn
Saskatchewan
O. Amici
Bari
H. Crapo
Montreal
S. Antonucci
Napo 1i
M. Crismale
Bari
F. Bachmann
Kiel
I. Debroey
Diepenbeek
A. Barlotti
Bologna
F. de Clerck
Gent
M. Barnabei
Bologna
C. De Concini
Pisa
Bartocci
Perugia
M. De Finis
Roma
C. Bartolone
Palermo
M. Dehon
Bruxelles
A. Basile
Perugia
A. Delandtsheer
Bruxelles
A. Basile
Roma
J. Denes
Budapest
L.M. Batten
Winnipeg
R.H.F. Denniston
Leicester
L. Beneteau
Toulouse
M.J. de Resmini
Roma
C. Bernasconi
Perugia
M. Deza
Paris
T. Beth
Erlangen
V. Dicuonzo
Roma
A. Beutel spacher
Mainz
B. d'Orgeval
Dijon
A. Bichara
Roma
J. Doyen
Bruxelles
M. Bil i otti
Leece
L. Dubikajtis
Katowice
P. Biscarini
Perugia
E.W. Ellers
Toronto
F. Bonetti
Bologna
H. Ell ers
Toronto
L. Borzacchini
Bari
G. Faina
Perugia
S. Bourn
Adelaide
S. Fanelli
L' Aquila
A.A. Bruen
Western Ontario
A. Farinola
Bari
T. Brylawski
Chape 1 Hi 11
M. Ferri
Napoli
F. Buekenhout
Bruxell es
O. Ferri
L' Aquil a
D. Buset
Bruxelles
J. Ch. Fischer
Regina, Canada
K.A. Bush
Bologna
M. Gionfriddo
Catania
F. Candori Conti
Perugia
T.S. Griggs
Preston
M. Capursi
Bari
T. Grundhofer
Ka i sers 1autern
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Ch. Hering
Tubingen
P.V. Ceccherini
Roma
R. Hi 11
Salford
M. Cerasoli
L'Aquila
J.W.P. Hirschfeld
Sussex
A.M. Cohen
Amsterdam
H. Hotje
Hannover
U.
822
List o[participants
X. Hubaut
Bruxelles
S.L Payne
Oxford, Ohio
D.R. Hughes
London
G. Pellegrino
Perugia
T. lhringer
Darmstadt
Bruxelles
N.L. Johnson
Iowa
Mil ano
W.M. Kantor
Murray Hill
N. Percsy C. Perelli Cippo R. Permutti
H. Karzel
MUnchen
M. Pertichino
Bari
A.D. Keedwell
Guildford
H. Pflugfelder
Philadel phia
G. Korchmaros
Budapest
S. Pianta
Brescia
H.J. Kroll
MUnchen
G. Pica
Napo 1i
B. Lara to
Bari
P. Plaumann
Erlangen
H. Leemans
Gent
L. Puccio
Messina
C. Lefevre Percsy
Bruxelles
C. Procesi
Roma
D.C. van Leijenhorst
Nijmegen
G. Quattrocchi
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D. Lenzi
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P. Quattrocchi
Modena
J. Lester
Hamburg
G. Raguso
Bari
M. Leuci
Bari
R. Rink
Kaiserslautern
M. Limbos
Bruxelles
M. Ronan
Chicago
J.H. van Lint
Eindhoven
A. Rosa
Hamilton
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L.A. Rosati
Firenze
L. Lombardo-Radice
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I.G. Rosenberg
Montreal
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G. Lunardon
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J. Saxl
Cambridge
H. LUneburg
Kaiserslautern
G. Selmi
Lecce
A. Machi M. Marchi H.F. Mattson jr. F. Mazzoeca G. Meniehetti
Roma
B. Simeone
Roma
Brescia
D. Smit Ghine11i
London
Syracuse
C. Sonrna
Roma
Napol i
R. Spanieciati
Roma
Bologna
K. Strambach
Erlangen
G. Miee11i
Leece
E. Strickland
Roma
W. Mielants
Gent
G. Tallini
Roma
G. Migl; or;
Roma
M. Tallini Scafati
Roma
S. Mil ie i
Catania
L. Teirlinck
Bruxelles
A. Neuen
London
J.A. Thas
Gent
D. Olanda
Napoli
M.J. Thomsen
Hamburg
A. Pas i ni
Siena
A. Torre
Pavia
Trieste
List ofparticipants
823
Bruxelles
M. Vi ncent-Wi 11 ems A. Ivic Weiss
Antwerpen Toronto
S.A. Vanstone
Waterloo
R. Weiss
Medford
K. Vedder
Giessen
K.E. Wolff
A. Venezia
Roma
J.C.D.S. Yaqub
Giessen TUbingen
A.G.S. Ventre
Napo 1i
R. Vincenti
Perugia
G. Zappa H. Zeitler
Firenze Tirshenrenth
E. Zizioli
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E. Ughi
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ANNALS OF DISCRETE MATHEMATICS Vol. I:
Studies in Integer Programming edited by P. L. HAMMER, E. L. JOHNSON, B. H. KORTE and G. L. NEMHAUSER 1977 viii + 562 pages
Vol. 2:
Algorithmic Aspects of Combinatorics edited by B. ALSPACH, P. HELL and D. 1. MILLER 1978 out of print
Vol. 3:
Advances in Graph Theory edited by B. BOLLOBAs 1978 viii + 296 pages
Vol. 4:
Discrete Optimization, Part I edited by P. L. HAMMER, E.L. JOHNSON and B. KORTE 1979 xii + 300 pages
Vol. 5:
Discrete Optimization, Part II edited by P. L. HAMMER, E.L. JOHNSON and B. KORTE 1979 vi + 454 pages
Vol. 6:
Combinatorial Mathematics, Optimal Designs and their Applications editedbyJ. SRIVASTAVA 1980 viii + 392 pages
Vol. 7: TopicsonSteinerSystems edited by C. C. LINDNER andA. ROSA 1980 x + 350 pages Vol. 8:
Combinatorics 79, Part I edited by M. DEZA and I. G. ROSENBERG 1980 xxii + 3 10 pages
Vol. 9:
Combinatorics 79, Part II edited by M. DEZA and I. G. ROSENBERG 1980 viii + 3 10 pages
Vol. 10: Linear and Combinatorial Optimization in Ordered Algebraic Structures edited by U. ZIMMERMANN 198 I x + 380 pages Vol. I I: Studies on Graphs and Discrete Programming edited by P. HANSEN 1981 viii + 396 pages
Vol. 12: Theory and Practice of Combinatorics edited by A. ROSA, G. SABIDUSI and J. TURGEON 1982 x + 266 pages Vol. 13: GraphTheory edited by B. BOLLOBAs 1982 viii + 204 pages Vol. 14: Combinatorial and Geometric Structures and their Applications edited by A. BARLOTTI 1982 viii + 292 pages Vol. 15: Algebraic and Geometric Combinatorics edited by E. MENDELSOHN 1982 xiv + 378 pages Vol. 16: Bonn Workshop on Combinatorial Optimization edited by A. BACHEM, M. GROTSCHELL and B. KORTE 1982 x + 312pages Vol. 17: Combinatorial Mathematics edited by C. BERGE, D. BRESSON, P. CAMION and F. STERBOUL 1983 x + 660 pages in preparation
