THEORY OF FINITE GROUPS A Symposium
RICHARD BRAVER Harvard University and
CHIH-HAN SAH University of Pennsylvania
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THEORY OF FINITE GROUPS A Symposium
RICHARD BRAVER Harvard University and
CHIH-HAN SAH University of Pennsylvania
o W. A. BENJAMIN, INC.
New York
1969
Amsterdam
: : :::ORY OF FINITE GROUPS , . ~·.;'osium
Copyright © 1969 by W. A. Benjamin, Inc . .-\11 rights reserved
Library of Congress Cata10g Card Number 69-20486 \tanufactured in the United States of America ;23.+5M32109 The manuscript was put into production on November 8, 1968; this volume was published on February 15, 1969
\\. A. Benjamin, Inc. "ew York, New York 10016
A Note from the Publisher This volume was printed directly from a typescript prepared by the editors, who take full responsibility for its content and appearance, The Publisher has not performed his usual functions of reviewing, editing, typesetting, and proofreading the material prior to publication. The Publisher fully endorses this informal and quick method of publishing conference proceedings, and he wishes to thank the editors for preparing the material for publication.
Preface These proceedings consist of abstracts of the lectures presented at a regional conference on finite groups held from May 2 to May 4, 1968, at Harvard University.
In addition, a number of
colleagues who were not able to lecture on their recent works at this conference were kind enough to submit abstracts to our collection.
We hope that
these papers may give a picture of some of the recent progress in this field.
We would like to acknowledge our appreciation to the financial assistance rendered by Harvard University, Massachusetts Institute of Technology and the National Science Foundation which made th conference possible.
We would also like to thank
W. A. Benjamin, Inc. for publishing these reports.
Finally, without the care and patience of Misses Taffy Jones and Mary Vallery, the manuscripts could never have been prepared for camera copy. R. B. C. H. S.
vii
TABLE OF CONTENTS
Part I Page
1.
On groups with quasi-dihedral Sylow 2-subgroups, I.
2.
21
odd.
=:
1
PSP4(q)
W. J. Wong
(mod 4)
, 31
A characterization of q
and
25
On a characterization of q
7.
H. Bender
P. Fong
D4 (q)
6.
13
On a characterization of 2
5.
R. Brauer
Finite groups having a strongly embedded subgroup.
4.
1
On groups with quasi-dihedral Sylow 2-subgroups, 11.
3.
J. L. Alperin
L (q) 4 K. W. Phan
for 39
Finite groups with abelian 2-Sylow groups.
J. H. Walter
ix
43
x
Contents
8.
A theorem on Jordan groups.
9.
On the alternating groups.
10.
N. Ito
47
T. Kondo
49
Uniprimitive permutation groups. L. L. Scott
11.
Some new simple groups of finite order.
z. 12.
Janko
63
On Janko's simple group of order 50,232,960.
13.
55
G. Higman and J. McKay
The simple group of order
604,800.
M· Hall, Jr. and D. Wales
14.
79
Le groupe de Janko d'ordre
604,800.
J. Tits
15.
91
Linear groups of degree Hall-Janko group.
16.
6
and the
J. H. Lindsey, 11
97
On the isomorphism of two groups of order
17.
65
44,352,000.
C. C. Sims
A simple group of order J. McLaughlin
101
898,128,000. 109
xi
Contents 18.
A simple group of order
448,345,497,600.
M. Suzuki
19.
113
Some simple groups related to
M . 24
Dieter Held
121
Part 11 1.
An analysis of group representations. S. B. Conlon
2.
125
Centralizer rings and characters of representations of finite groups. W. Curtis and T
3.
131
Some properties of the Green correspondence.
4.
V. Fossum
W. Feit
139
1ndecomposable modules for finite groups. G. J. Janusz
5.
Relative Grothendieck group. and I. Reiner
149
T. Y. Lam 163
xii 0.
Contents Isometries and characters of finite groups. W. F. Reynolds
7.
171
On the integral representation of an order.
H. Zassenhaus
181
Part III 1.
The order of finite Chevalley and Steinberg group.
2.
N. Iwahori
203
The Steinberg character of a finite group with a BN pair.
4.
L. Solomon
223
On the centralizer of involutions in finite groups.
6.
213
A sufficient condition for p-stability. G. Glauberman
5.
195
On some properties of groups with BN-pairs.
3.
R. Carter
D. Gorenstein
227
Centralizer of involutions in finite simple groups.
On F-projectors.
J. H. Walter
239
B. Huppert
243
Contents 8.
Locally finite versus finite simple groups.
9.
la.
xiii
O. H. Kegel
Automorphic Algebras.
247
Ernest Shult
251
Automorphisms of finite groups. C. H. Sah
261
NOTE: Pages 158-162 are not missing from this publication. Pages were numbered incorrectly when the final manuscript was produced.
ON GROUPS WITH QUASI-DIHEDRAL SYLOW 2-SUBGROUPS, I
J. L. Alperin
Our object is the ultimate classification of another class of simple groups, namely, those with a Sylow 2-subgroup
S
defined by generators
a
2
n
of order a, b
b -1
+ 1, n ~ 3,
2
and relations
+
2
n
-
1
The only simple groups known with such quasidihedral Sylow 2-subgroups are q
~
-1
(mod 4), U (q) 3 (The group
linear group
L (q) 3
with
q
L (q) 3
=1
with
(mod 4)
and
is the projective special
PSL(3, q), the group
projective special unitary group 1
U (q) 3 PSU(3, q)
is the and
FINITE GROUPS
2
M ll
is the Mathieu group on eleven letters.)
The
main result is a major step towards a classification of all simple groups with quasi-dihedral Sylow 2-subgroups.
This work has been done jointly
with R. Brauer and D. Gorenstein; the final draft is in preparation and is still subject to change.
MAIN THEOREM.
Let
G
be
~
simple group
with quasi-dihedral Sylow 2-subgroup and let be an involution in
-
either
C(x)
is isomorphic to
GL(2, q)
GU(2, q) with
(i i)
If
either
q
3
and
with
q -
~
guotient of
q = -1 (mod 4)
q - 1 (mod 4)
group of odd order
or
G .
-
(i)
x
£y
~
or
central sub-
d .
c
(mod 4),
6
+ 1
then
3
IGI =
7920 •
This result is very close to a classification. In fact,
if
q
;:
-1
(mod 4)
then this theorem and
a previous one of Brauer [1] isomorphic to d
=
(3, q
-
L (q) 3
1) .
or
imply that
is
G
and that
M
n
The case that
-
q
1
(mod 4)
is
still open but it is possible one may show that G
is a doubly transitive group of a special sort.
Hopefully, one may also prove that 3
(i t
should be
(3,
q + 1)
).
d
is
1
or
If this can be
accomplished then a result of Suzuki [3] may be applied to handle the case that
d
=
1 .
The proof of the Main Theorem is quite long and divides into three major stages.
The methods
are local group-theoretic and character-theoretic. We shall concentrate here on the former: for the latter the reader should see the paper by Brauer in this collection. these three parts.
We shall now survey each of
4 1.
FINITE GROUPS General results Various previous classification theorems
allow us to give detailed results on the structure of subgroups of groups Sylow 2-subgroups. x N
G
For example, if
is an involution of
= N/<x)
then
with quasi-dihedral
N
G
G, N = C(x)
is simple,
and
has dihedral Sylow 2-subgroups
so the results of Gorenstein-Walter [2J can be applied.
This yields that
has a series of
N
normal subgroups
N
::J
L
::J
L
o
::J
S (N)
::J
°
(N)
::J
1
with the following properties: (i)
O(N)
odd order in (ii)
N
S(N)
(iii) for some odd (iv)
is the largest normal subgroup of
=
q2
number
and that p.
q2
q2
are
Suppose that
is a power of the prime
We apply methods and results of
Brauer [2,3J
(on groups whose Sylow 2-subgroups
are quasi-dihedral or the wreath product of a cyclic group and a group of order 2) to centralizers in
G
of certain elements of odd order, in
order to show that a Sylow p-subgroup of is not one of
G.
a certain section
C(t)
This implies the existence of V
in
G
Abelian p-group and on which
which is an elementary
L
l
acts non-trivially.
The degrees of the irreducible Brauer characters of
for the prime
p
being known, we find a
lower bound for the dimension of
V
as a vector
35
space over the field of
p
elements.
On the
other hand, there is a four-group acting on
V
and we can find an upper bound for the dimension. The two bounds are compatible only when q2 = 3.
Then
V
L
l
5 ,
is the Frattini factor group of 3
an extra-special group of order action of
=
ql
on
V
5
,
and the
corresponds to an embedding in such a way that an
element of order
3
leaves the vectors of a
hyperbolic plane fixed. such embedding of
However, there is no
SL (5) 2
This
contradiction shows that
and
must be
powers of the same prime number. The identification of
G
with
PSP4(Ql)
in
case (ii) was obtained by Janko in the case Ql
=
3
[6J.
general case.
Similar methods can be used in the First we determine the structure of
the centralizer in gate to
t
G
of an involution not conju-
This is done by transfer methods, a
theorem of Gorenstein and Walter on groups with dihedral Sylow 2-subgroups [5J, and a
formula
FINITE GROUPS
36
of Brauer on the numbers of fixed points of elements of a four-group acting on a group of odd order [1].
We find that the centralizer is a
split extension of a normal subgroup isomorphic with
PSL (ql) 2
2(ql - €)
by
where
a dihedral group of order ql
=0
(mod 4),
€
Having done this, we set up a for
G
?-subgroup of ?rime number
(BN)-pair
in
G
G, where
ql
is a power of the
p.
of a Sylow
This is done by determining
centralizers and normalizers of various smaller p-subgroups of
C(t)
main tool is the Brauer formula referred to
320ve. a~d
±l .
B
?-subgroups, beginning with ~~e
=
Most of the work is involved in deter-
Dining the normalizer
~~e
€
Because of the abundance of four-groups
particularly the existence of two classes of
~~volutions,
we do not have to use directly any
arguments from the theory of group characters. ~~entually ~or~alizer
s~o·.,:
we determine the exact structure of the B
of a Sylow p-subgroup of
G
that together with a certain subgroup
and N
of
37 C (t)
it forms a
then
GO
(BN)-pair [7].
is a subgroup of
order of
G
If
=
GO
BNB ,
whose order is the
and the structure of
uniquely determined. Since
is
It follows that GO
has two classes of
involutions, and contains the centralizer in of each of its involutions, we have
GO
=
G .
G
FINITE GROUPS
38
REFERENCES
1.
R. Brauer, Some applications of the theory of blocks of characters of finite groups 11, J. Algebra 1 (1964), 307-334.
2.
R. Brauer, On finite Desarguesian planes 11, Math. Z. 91 (1966), 124-151.
3.
R. Brauer, Investigations on groups of even order 11, Proc. Nat. Acad. Sci. USA 55 (1966), 254-259.
4.
G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403-420.
5.
D. Gorenstein and J. H. Walter, On finite groups with dihedral Sylow 2-subgroups, Illinois J. Math. 6 (1962), 553-593.
6.
Z. Janko, A characterization of the finite simple group PSP4(3), Canad. J. Math. 19 (1967), 872-894.
7.
J. Tits, Theoreme de Bruhat et sous-groupes paraboliques, C. R. Acad. Sci. Paris 254 (1962), 2910-2912.
A CHARACTERIZATION OF FOR
K.
q
w. Phan,
'" -1
(mod 4)
Bonn (Germany)
The following result has been proved: THEOREM:
Let
t
o
be an involution contained L (q) 4
in the center of a Sylow 2-subgroup of where
q
H
o
Let
is congruent to the centralizer of G
be
~
-1 t
4.
modulo o
Denote
in
finite group with the following
properties: (a)
G
has no subgroup of index 2,
G
has an involution
and, (b)
that the centralizer of
t
in
G
t
C (t) = H G
is isomorphic to 39
such
H
o
FINITE GROUPS
40 Then
G
is isomorphic to
L (q) 4
The methods of the proof are group-theoretic. Our main aim is to construct two subgroups of fulfilling the properties of a sense of Tits.
G
(B,N) pair in the
Later we show that these two sub-
groups generate a subgroup
G o
of
and finally prove that
isomorphic
G
G
o
=
G .
To
this end we begin with the study of the structure of
H, in particular the determination of the
number of classes of involutions in centralizers.
H
and their
Straight-forward arguments and a
result of Thompson show among other things that G
has two classes of involutions and is a simple
group. H
The manner in which the involutions of
fuse together in
G, the first theorem of
Grun and a result of Gorenstein-Walter are used to determine the structure of the centralizer of an involution not conjugate to
t
in
G .
From the structure of the centralizer of an involution of the second type we obtain a series of results on
G
It is then possible to
41 determine the structure of a Sy10w p-subgroup of G
and its normalizer in
divisor of
q.
where
p
is the
The Sylow p-normalizer
the normalizer
N
of a complement in
Sy10w p-subgroup of G
G
B
mentioned earlier.
Band B
to the
are the two subgroups of With the structure of these
two subgroups and the action of some elements of N
on some subgroups of
to show that
BNB
~
B
L (q) 4
known, where
BNB
union of the set of double cosets in
N
that
it is then easy
BnB
is the with
n
Applying a result of Thompson shows
=
BNB
G
completing the proof.
We hope to show that the theorem is true for q
=
4h + 1
where
h
is an integer.
Some of the
main results obtained so far are: 1.
The centralizer
H
of
t
in
G
is a
non-splitting extension of the central product of two subgroups isomorphic to hedral group of order of
q
= 2.
-1
(mod 4)
When
q
=5
SL(2, q)
2(q - 1).
by a di-
(In the case
the extension splits). (mod 8)
G
has one class of
42
FINITE GROUPS
involutions.
that
3.
The group
4.
A Sylow p-subgroup of
5.
There exists a subgroup
NG(K)/K
G
is simple. G
has order K
of
G
q
6
such
is isomorphic to the symmetric
group on 4 letters. A similar characterization of
U (3) 4
has
been done and will appear in the Journal of the Australian Math. Society.
It is hoped that the
methods can be generalized to the case of wi th odd
q
U (q) 4
FINITE GROUPS WITH ABELIAN SYLOW 2-SUBGROUPS
J. H. Walter
The purpose of this note is to announce a result obtained last year but not yet published. Let
G
be a finite group.
Denote by
O(G)
maximal subgroup of odd order.
Denote by
the minimal normal subgroup
of
G/N
N
G
the 0' (G)
for which
has odd order. THEOREM.
abelian
Let
G
S2-subgroups.
direct product of
~
be
~
finite group with
Then
0' (G/O(G))
is the
2-group and simple groups of
the following types: n 2 ) ,
(1)
PSL(2,
(2)
PSL(2, q)
q
>
3
n
>
, q - 3
,
43
1 or
5 (mod 8)
,
44
FINITE GROUPS (3 )
~
simple group
involution
J
where
K
or
(mod 8)
5
of
M
such that - -for - -each --
M
is isomorphic to
PSL(2, q) , q - 3
Simple groups of the third type have not yet been completely determined.
However,
it is known
from the results of Janko and Thompson [3] that q =
5
or
q = 3
2n + 1
, n
>
In the former
0 .
case, the group is the simple group of order 175,560 discovered by Janko [2].
In the latter case the
groups are presumably the Ree groups associated with the Lie algebra istic 3.
G
2
over a field of character-
John Thompson [5] has made considerable
progress toward showing that this is the case. Their character tables have been worked out by H.N. Ward [6]. where
q
__ 32n
Their order is
q
3
(q
3
+
1) (q -
1)
+ 1
The groups of the first type are characterized in terms of conditions given by Suzuki [4] The groups of the second type are groups with a four group as an s2-subgroup and hence are
,
45 characterized by Gorenstein and Walter [lJ.
REFERENCES
1.
Gorenstein, D. and Walter, J. H. The characterization of finite groups with dihedral Sylow 2-subgroups. J. Algebra 2 (1965), 85-151, 218-270, 354-393.
2.
Janko, Z. A new finite simple group with abelian 2-Sylow subgroups and its characterization. J. Algebra 3 (1966), 147-186.
3.
Janko, Z. and J. G. Thompson. On a class of J. Algebra 4 finite simple groups of Ree. (1966), 274-292.
4.
Suzuki, M. Finite groups of even order in which the Sylow 2-subgroups are independent Ann. Math. 80 (1964), 58-77.
5.
Thompson, J. G. Towards a characterization of E *(q), J. Algebra 7 (1967),406-414. 2
6.
Ward, H. N. On Ree's series of simple groups. Trans. Am. Math. Soc. 121 (1966), 62-89.
A THEOREM ON JORDAN GROUPS
Noboru Ita
Let
G
be the set of symbols
> 3)
1,2, ... ,n
called points.
Let
G
doubly transitive permutation group on that the stabilizer and
2
3, ... , k
in
G
G l
~
,2
such
of the points 1
stabilizes further points
~3;
is
(mod k) k
=
2
,
,
q n
where
=
q
2
q and
is a power G
is
isomorphic to the split extension of the 2-dimensional vector space field
GF (q)
and
7
(mod k)
faithful, then
(3 )
then
1 ,
of
q
V (2, q)
elements by
over the
GL (2, q)
ON THE ALTERNATING GROUPS
Takeshi Kondo
This is a summary of the forthcoming papers [5] and [6], which will be published in Jour. of Soc. of Japan.
M~th.
I.
Lpt
letters
(1,
where
n
Define
A
m
2,
be the alternating group on
... ,
m} •
Put
m
=
4n + r
is a positive integer and n
involutions
a
k
(1
~
k
~
0
n)
, r
~
m
of
3
~
A
m
as follows: a
Then
=
k
a
n
(1,
(4k-3, 4k-2) (4k-l,
4)
4k).
is contained in the cent er of a 2-Sylow
subgroup of note by
2) (3,
A
m
H(n, r)
For
r
=
1 ,
2 and 3 , we de-
the centralizer in
Then we have the following theorems.
49
A m
of
a
n
FINITE GROUPS
50
Theorem 1.
Let
G(n, r)
be a finite group
with the following properties: (1)
G(n, r)
has no subgroup of index 2,
G (n,
contains an involution
and ( 2)
r)
in the center of G (n,
r)
~
Then if to r
=
A
4n + r
2
where
r
=
2
or
2)
For the case
r
C
G(n, r)
(a )
n
H(n, r)
3, G (n, r)
except for the G(l,
n
2-Sylow subgroup of
whose centralizer
is isomorphic to
a
~ A
=
or
6
n
~
is isomorphic 1
and
PSL(2, 7)
1 , we have not obtained
the analogous result, but we can prove much weaker result.
We note that
H(n, 1)
elementary abelian subgroup
has a unique
S
of order
22n
up
to conjugacy. Theorem 2.
Let
G(n, 1)
be a finite group
contcining an involution whose centralizer ~
isomorphic to
H(n, 1)
abelian subgroup of
, and
H(n, 1)
S
H(n, 1)
be an elementary
of order 22n .
51 ~ ~-to-one
Assume that there exists from in
e
mapping
8
U N (S) (the set-theoretic union A m such that onto H(n, 1) U N G(n, 1) (S)
H(n, 1)
A ) m
induces an isomorphism
(resp.
(S»)
N
A
Then
and
G(n, 1)
1)
(resp. NG(n, l)(S))
is isomorphic to
(H. Yamaki)
G(3, r)
Th. 1.
H (n,
H(n, 1)
m
Theorem 3. group
between
Let
or
G
be
r
~
finite
satisfying the assumption of
Then (i)
if
r = 1 or
G
is isomorphic
l
G
has precisely four
l
conjugacy classes of involutions, and (ii) if r
=
2
or
n.
3,
G
r
Remarks.
is isomorphic to (1 )
If
r
=
Yamaki showed that
G l
A 12
r
Th. 3 1
2.
In fact,
has two possibilities
for the fusion of involutions of which is that of
+ r
12
2 or 3
is a special case of Th. 1, but if Th. 3 is a better result than Th.
A
or
A
that of the symplectic group
l3
G l
,
one of
and the other SP6(2)
1.5
(Note that
satisfies the condition of Th. 3 for
52
r
FINITE GROUPS
=
if
1.)
Then it is not difficult to see that, has the first case for the fusion of in-
G 1
volutions, (2)
G
l
satisfies the assumption of Th.
2.
The work of Yamaki was done before
Th. 1 and Th.
2 were obtained.
In fact,
a part of
the proof of Th. 1 and 2 owes to his idea. (3)
In [4; Th. A], we determined the fusion
of involutions of
G(n, r)
for the case
r
=
2 or 3.
This is equivalent to the determination of the structure of
NG(n, r) (S)
,where
mentary abelian subgroup of (If
r = 2 or 3 ,
to conjugacy.)
S
H(n, r)
is some eleof order 2
2n
such subgroup is not unique up
But we have not obtained a result
similar to [4; Th. A]
for the case
r = 1.
This
is the reason why the stronger condition is necessary for Th. t~at,
of
2.
However, we note that Th.
2 shows
if We can determine the fusion of involutions
G(n, 1)
under the same assumption as Th. 1,
we shall be ?ble to obt?in a similar result to T~
1 also for the case (4)
r = 1 .
For small value of
m,
Th. 1 and
53
Th. 2 were treated by A. Fowler, M. Suzuki and D. Held.
We used these results for the proof of
our theorem. In [3], we proved the following theorem
Ill.
which is a generalization of W. J. Wong's theorem
Theorem.
Let
G
be a finite group satisfy-
ing the following conditionr there exist volutions
a 1'
one mapping
a
2
,
... , n
from
CO
a
U
k=l theoretic union in that and
A )
m
of
n CA
G
(~k)
m onto
and a ---
isomorphic to
A
~ n)
Then if
in-
one-to-
(the setn U CG (a k ) k=l
induces an isomorphism between CG (a ) (1 < k k
n
m
>8 =
C
A m
such (a~)
k
G
is
with
A m
,
m
For the identification of
G(n, r)
we used this theorem, the proof of which is due to the idea of D. Held [1] and [2].
54
FINITE GROUPS REFERENCES
1.
D. Held, A characterization of the alternating groups of degree eight and nine, J. of Alg., 7 (1967), 218-237.
2.
-----, A characterization of some multiply transitive permutation groups I, Illinois J. (to appear).
3.
T. Kondo, On the alternating groups, J. Fac. Sci. Univ. Tokyo (to appear).
4.
-----, On finite groups with a 2-Sylow subgroup isomorphic to that of the symmetric group of degree 4n , J. Math. Soc. Japan (to appear).
5.
-----, On the alternating groups II, appear) .
6.
H. Yamaki, A characterization of the alternating groups of degrees 12, 13, 14, 15 (to appear) .
(to
UNIPRIMITIVE PERMUTATION GROUPS *
Leonard L. Scott, Jr.
A well-known result of Wielandt (lOJ is Theorem group on 2p
~
2
m
(Wielandt) 2p
+ 1
Let
letters,
be a uniprimitive
@
p
a prime.
Then
for some odd positive integer
m.
Despite the specific nature and simplicity of the condition
2p
~
2
m
+ 1 , the only known
examples of uniprimitive groups of degree p 6
a prime, occur when
5
p
~
5
acting on the 2-subsets of (1,
m
~
3
2p, ~5
and
2, 3, 4, 5) ).
The term " un iprimitive" is due to W.A.Manning * and refers to primitive permutation groups which are not doubly transitive. The results announced in this article are part of the author's doctoral dissertation submitted to the faculty of Yale University. 55
56
FINITE GROUPS
For a given prime
p, the existence of a group
which is uniprimitive of degree 2p
implies the
existence of a simple group with the same property. Wielandt showed in unpublished work that no uniprimitive group of degree despite
= 52 + 1 .
26
=
2.13
26
exist,
The author has proved that,
in fact, If
Theorem 1 2p
=
p
2 m + 1
5
m
, =
@
is a uniprimitive group of degree
where
p
and
m
3 (and
@
= m5
or
are primes, then 15
5
Feit has recently shown (unpublished) that the case
p = 41
eliminated
does not occur, and the author has p = 113.
tive groups of degree unless
p
=
Thus there are no uniprimi2p
p
=
113
p
p
2 m + 12
such that
7
2 m + 15
or
to be the set of primes
which satisfy at least
f
A)
If
or
3p - 1 , then
B)
4
q
2
48p
.
p - 2 (mod 3 )
of A), B) , C) below:
~
is a prime divisor of q
Define
~
1 5
-
~
"4 3 p
is the exact power of
p - 1
+110
2
dividing
61
and if
p - 1
P - 1
of
(mod 3) C)
=
P
or or
f
q
2
3p q
1 + 2q
1 , then either
FP
1 5
~
is a prime divisor -
f
2
1
+ 110
1 + 4q
or
q
where
q
is
a prime. Theorem 2
If
P
is a prime in
uniprimitive group P E n* acter
and 8
@
@
on
3p
n* n n**
then no
letters exists.
If
exists, then the permutation char-
is a sum of irreducible complex characters
the characters distinct, real-valued,
are
and algebraically conjugate.
The proof uses the theory of characters of small degree [2], [3], the modular theory, of the centralizer ring.
and the theory
FINITE GROUPS
62
LITERATURE
1.
Brauer, R. "On groups whose order contains a pr ir..e to the fir st power," 1,11. Art:. J. Math. 64 (19"+2)
2.
Feit, ','1. "Groups with a cyclic Sylow subgroup." Nagoya Math. J. 27 (1966).
3.
Feit, W. "On finite linear groups." Algebra 5 (1967).
4.
Green, J.A. "A transfer theorem for modular representations." J. Algebra 1 (1964).
5.
Higman, D.G. "Finite permutation groups of rank 3." Math. Z. 86 (1964).
6.
Ito, N. "On uniprimitive groups of degree 2p." Math. Z. 78 (1962).
7.
Scott, L. "Uniprimitive groups of degree kp." Doctoral dissertation, Dept. of Math. Yale U. 1968.
8.
Sims, C.C. "Graphs and finite permutation groups." Math. Z. 97 (1967).
9.
Tamaschke, O. "A generalized character theory on finite groups." Proc. Internat. Conf. Theory of Groups, Austral. Nat. Univ., Canberra, August 1965, pp. 347-355. Gordon and Breed Science Publishers, Inc. 1967.
J.
10.
Wielandt, H. "Primitive Permutationsgruppen von Grad 2p." Math. Z. 63 (1956).
11.
Wielandt, H. Finite Permutation Groups. Academic Press 1964, New York.
SOME NEW SIMPLE GROUPS OF FINITE ORDER
Zvonimir Janko (Monash University, Melbourne)
We prove the following result: Theorem.
Let
G
be a non-abelian finite
simple group with the following properties: (i) T
of
G (ii)
The cent er
Z(T)
of a Sylow 2-subgroup
is cyclic. If
z
is the involution in
then the centralizer tension of a group
H E
of
z
of order
in 32
G by
Z(T) is an exAS •
We then have the following possibilities. G
has only one class of involutions, then
order
50,232,960
character table.
G
If has
and a uniquely determined If
G
has more than one class 63
64
FINITE GROUPS
of involution, then
G
has order
604,800
and a
uniquely determined character table.
The existence of a simple group of order 50,232,960
was shown by Graham Higman and J. McKay.
By a result of S. K. Wong follows that any simple group of that order must satisfy the conditions (i)
and (ii) of our theorem.
Also John G. Thompson
has shown that a simple group of that order must possess a subgroup isomorphic to
PSL(2, 16)
and
so using this fact the work of Graham Higman then shows that such a group is unique. The existence of a simple group of order 604,800 was shown at first by M. Hall, Jr.
Also a
very elegant geometric construction of this group was obtained by J. Tits.
It was proved by D. Wales
that there exists only one simple group of order 604,800.
ON JANKO'S SIMPLE GROUP OF ORDER
50,232,960
Graham Higman and J. McKay
1.
INTRODUCTION Z. Janko in [1] considered a finite simple
group
G (i)
such that G
contains an involution whose
centraliser is an extension of an extraspecial group of order
32
by
SL(2, 4)
In particular, he showed that if furthermore (ii) then
G
all involutions in has order
50,232,960
character table given in [1].
G
are conjugate
and has the He left open the
question of the existence and uniqueness of such a group.
Here we report briefly on work, partly
group-theoretic and partly computational, which shows that a group does exist satisfying (i), 65
(ii),
66
FINITE GROUPS
and also G
(iii)
has a sUbgroup
extension of
SL(2, 16)
morphism of order Conditions (i)
H
which is the
by an outer auto-
2.
to (iii) determine
isomorphism, and
G
G
up to
has an outer automorphism
extending the outer automorphism of
H
(i.e.
extending the automorphism of order
4
of
SL(2, 16)
2.
).
PERMUTATION CHARACTERS The first step was a systematic search,
carried out on the computer, for characters of
G
which might be permutation characters corresponding to large subgroups. character (a)
Necessary conditions for a
~
to be a permutation character are
~
contains the principal character just
once; (b)
~
(c)
~(x)
for all (d)
~
divides
(1)
(1)
IGI
is a non-negative rational integer x
in
G;
divides
hi ~ (x)
to a class containing
h. l
, if
x
belongs
elements; and
67
(e)
for all
all positive integers ~
By (c),
x
in
G
and
k. conjuga~e
contains algebraically
characters with the same multiplicity, so we add these together to obtain the rational table of to the of
G , with 14
G.
14
firs~
characte~
characters, corresponding
conjugate classes of cyclic subgroups
Then we generate the
2
13
combinations
of these characters which satisfy (a) all multiplicities are for conditions (b)
0
or
to (e).
and in which
1 , and test them
This takes
6
seconds.
Besides the principal character, there emerge characters of degrees
6156, 19380
and
20520,
corresponding, possibly, to subgroups of orders 8160, 2592 be subgroups
and H
2448.
The first of these could
as in (iii)
be subgroups isomorphic to
above; the last could PSL(2, 17)
Repeating the process, but allowing also multiplicity
2
takes
7
minutes.
Of the
characters that emerge, some can be seen to correspond to subgroups known to be present (if
G
exists at all), such as centralizers of involutions, and others can be eliminated.
But there are also
FINITE GROUPS
68
characters which might correspond to subgroups of whose presence we cannot be certain, in particular. subgroups isomorphic to
PSL(2, 19)
Finally, using a different program, assisted by the fact that
G
has only one non-principal
character of odd degree, the factors of
IGI
were
examined in increasing order of magnitude, with no bound on the multiplicities, except the natural one that the mUltiplicity of an absolutely irreducible character shall not exceed its degree. done up to degree
This was
17442; two new possible degrees
emerged, including
12312, corresponding,
presumably, to the
SL(2, 16)
subgroup of index
3.
2
in
contained as a
H .
GENERATORS AND RELATIONS At this stage, it seemed best to assume the
existence of the subgroup
H , and see what
consequences this would have. We denote elements of
GF(16)
by
i 2 w oj , where w + w + 1 = 0 , so that 2 and 0 + wo + 1 so that 0 5 = 1 0 of
SL(2, 16 )
are matrices
(: ~)
with
0
and
3 w = 1 , Elements a, b, c, d
69
in
GF(16)
and
ad - bc
<SL(2, 16) , u>
group is
(::
2 b 2)
d
<SL(2, 16), u 2 >
normaliser GL(2,
4)
,
L
in
V
u
Then
H
.
of
of order
G
4
where
(1 *1)
The elements abelian subgroup
1 ; and its automorphism
=
=
1
and
is
form an elementary
H
16, whose
must be a split extension by
acting naturally (from a consideration
of the structure of the centraliser of an involution). But
N (V) H
is only of order
480
Thus
G , if
it exists, can be generated by an amalgam where
H
U L ,
For reasons which will be
H" L
mentioned in a moment, it seems best to consider not only
G , but also an extension
G l
of
G
by
an outer automorphism extending the automorphism of H
induced by
amalgam L H
l l
u
HI \J L l
To this end, we form first an where
is the extension of f'\ L
l
=
HI
= <SL(2, 16), u>
V
by
fL(2, 4)
and
, with
N (V) Hl
It can be verified that this amalgam is generated by the relations
Hl
and an element
t , subject to
70 (A)
FINITE GROUPS t
2
= 1
t-1C :)t = t-lr W 2)
t-lut
t
u
e
~)
(w
t -1
(1
(1
:) t
w2 0 2) 1
2)
W
3
Further relations, which must hold in
G,
though not in the free product generated by the amalgam, follow by considering the normaliser of the cyclic group generated by
(w
w2 ) .
The
structure of this normaliser follows easily from
[1]; it is a split extension of , where elements in and other elements of normalisers of
»
w 2
«W
PSL(2, 9)
centralise
PGL(2, 9) in
by
H
invert and
L
can be identified with subgroups of this extension in essentially only one way, and this leads to the relations
71
[t (1 1)] [t(p4 P)t(p2 p7f
(B)
u
4
2
1 .
Using relations (A) and (B) it is fairly easy to see that the element
transforms one subgroup of
SL(2, 16)
of order
34
into another, and hence to find an element not in H
which normalises a cyclic subgroup of
order
17
H
of
Drawing once more on [1] for the
structure of a Sylow 17-normaliser, this gives another relation, which simplifies to
( C)
1.
Thus if SL(2, 16)
G u
2
implied by (A),
exists, it is generated by and
t , subject to the relations
(B) and (C), and possibly others.
However, a little experimentation suggests that the relations we have are already sufficient.
If
this is so, then, since the relations we have are implied by (i),
(ii) and (iii), there can be at
most one simple group
G
satisfying these conditions.
FINITE GROUPS
72
Furthermore, if such a group exists, any two embeddings in it of the amalgam equiv~lent
HV L
under its automorphism group.
are In
partir.ular, the automorphism induced by conjugation by
u
in
H \J L l l
extends to an automorphism of
G , necessarily outer, because otherwise the Sylow 17-normaliser of
4.
G
would be too large.
COSET ENUMERATION
To check that we have indeed enough relations, and that these relations do not imply total collapse, we return to the computer, and carry out an enumeration of the cosets of
H, using a program
based on methods described by J. Leech [2J.
For
this purpose, we have to replace the matrix notation for
SL(2, 16)
and relations.
used above by one using generators Since the critical factor in
attempting the enumeration is the size of the immediate access store available, and we store the effect on each coset of each generator and its inverse, it is advantageous to minimise the sum of the number of involutory generators and twice the number of non-involutory generators.
73 This part of the work was in fact done twice, simultaneously and independently, by one of us (J. McK.) at Chilton and by M. J. T. GUy at Cambridge, using different programs and slightly different relations.
In both cases the answer
produced was the hoped-for one, that cosets in
G c
s2
(ac) =
(sa)2
b 5 tb- 5 t
=
where t
-2
a, b, c
ab
-3
3
= (bc) 2
(sc)2
= sbsb- 4
(b-2ctb4ct) 2 b
has
6156
The relations used by Guy were
2
t2
H
=
=
abab
(at)2 (ct)4 s
-4
ab
(bt)3
=
b 2 tb- l abtb- 2 a
=
u
completed in 77 seconds.
=
= =
is as in section 3.
1
= (b 2 st)3
2 3 3 7 4 ctab ctb ab ctactb ab ct
generate
3
SL(2, 16)
s
1 , 2
,and
His enumeration was No attempt has been made
to tidy up these relations, and it is, in fact, known that some of them are redundant.
However, to
delete redundant relations might well increase the running time, and aggravate the danger of generating so many redundant cosets that one runs out of store. A further coset enumeration, this time on
FINITE GROUPS
74
cosets of the subgroup
t~e
, shows that
t~ese
two elements already generate
Jo
PROPERTIES OF
G
G
What the coset enumeration shows, of course, SL(2, 16)
t~at
:5
~e:ations
crder ~~G
(A),
u
subject to the
Evidently
Both
groups on
6156
G
l
letters.
~aving
suborbit lengths
2040,
and
2720 , and
suborbit of length length
1360
G
for
G.
G,
G
G
l
of
G
u
2
of order
are permutation is of rank
7 ,
1, 85, 120, 510, 680, G
2720
are primitive. subgroup of
~ormal
and
G l
SL(2, 16)
generate a normal subgroup
50,232,960
and
t
(B) and (C) generate a group
2 x 50,232,960. t
and
is of rank
8 , the
splitting into two of It follows that both So, if N
N
G
l
is a minimal
is transitive, and
since there is no characteristically simple group
0:
order
s~bgroup
relations u
2
6156, of
H
(B),
N (\ H Thus
(C)
and so
is a nontrivial normal N
contains
then show that N = G
That is,
SL(2, 16) N
contains G
, and t
is simple.
There are two classes of involutions in
H ,
75 255
(1 1),
conjugates of
order
32 , an d
68
with centralisers of
' conJugates
'h u 2 , Wlt
0 f
centralisers of order
By
conjugate in
, and hence to
G
to
is
(B)
In the permutation representation of degree U
2
has
76
f'lXe d pOln ' t s.
whose stabiliser is fixed by
If
H , and
a
Then
6g
=
and so is conjugate in
H
g
is another point
(including
a
, and for
itself) b
-1 2
u g
a
g
of
belongs to
either to
Suppose that for
2
6
is the point
u 2 , then there is an element
such that
u
a
6156,
u
2
H
or to
choices of
-1 2 g u g
G
6
is conjugate to
choices conjugate to
(1 1).
Then a + b
=
32 , we have
76
a
=
16,
b
=
60 ,
1920 That is, to be satisfied.
has the right order for (i) To show that it has the right
structure, we need only recall three facts.
First,
76
FINITE GROUPS
2 CH(u ) so
contains a subgroup isomorphic to
C (u 2 ) G
is not soluble.
Second,
contains a sUbgroup of order element of order on it.
3
~)
CL (( 1
)
normalised by an
which acts fixed-point-freely
Thus there is a composition factor of
2
on which SL(2, 4) '" So CG(u 2 ) so faithfully.
acts non-trivially, and
C r (u )
group of order by
16
SL(2, 4)
2
C (u) G
(B),
namely
32
t
(1 1)
abelian.
Hence
=
E·SL(2, 4) 2
normal in
CG(u)
E
Finally,
contains an element of order , so that G
E
a
8,
cannot be elementary
satisfies
(i), and hence also
(ii) . The computing described in this note was done on the Atlas computers at Chilton and Cambridge, which have a cycle time of done while one of us
2 f4
sec.
The work was
(J. McK.) held a Science
Research Council research fellowship at Chilton.
REFERENCES 1.
Z. Janko, Some new simple groups of finite order I
2.
(to appear) .
J. Leech, Coset enumeration, article in Computational Problems in Abstract Algebra,
77 Pergamon Press, 1968.
The Mathematical Institute 24-29 St Giles Oxford The Atlas Laboratory Chilton Berks
THE SIMPLE GROUP OF ORDER
604,800*
Marshall Hall, Jr. and David Wales
Z. Janko [3J has characterized a simple group G
in terms of the centralizer of an involution in
the center of a 2-Sylow group the center
Z
of
P
E
fully on
32
of order E
If
involutions then
G G
He assumes that
is cyclic and that the central-
izer of the involution in group
P.
by
Z
AS
is an extension of a
,
which acts fai th-
has two conjugate classes of has order
604,800
and a
uniquely determined character table. Proceeding independently of Janko, we have shown [2J the existence and uniqueness of a simple *This research was supported in part by ONR contract N00014-67-A00094-00l0. 79
FINITE GROUPS
80 group of order
604,800.
The degrees of the
characters in the principal 7-block,
B (7) , can O
be determined using techniques described in [1]. ~\
7-Sylow group is its own centralizer and its
normalizer is of order c~aracters
in
B (7) O
a 5-block of defect 1
From the degrees of
42
it follows that 1
G
possesses
and two 3-blocks of defect
The centralizers and normalizers of these
jefect groups can be determined from the order of G.
Then the degrees of the characters in these
blocks and a number of the conjugate classes in G
can be found.
It is then possible to see that
G
possesses a 2-block of defect
mentary Abelian defect group.
2
with an ele-
This provides two
more degrees of characters, and using the orthogonality relations, the character table can be completed.
The table is given at the end of this
paper. The next step is to show that s~bgroup
of order
3-Sylow group
P3
6048
G
contains a
The normalizer
is of order
216
and if
N
of a R
is
81
4
an element of order of order
N n C
and
96
in
N, then
C
is of order
= C(R)
is
These
24
three groups can be determined specifically from the character table.
By examining the multiplicity
of the trivial character in the restrictions it can be shown that the group generated by group. H
Call it
is of order
group
H
6048
Nand
C
is a proper sub-
It then is easy to see that and isomorphic to the unitary
D (3) 3
We can now show the existence and uniqueness of
G
as a permutation group on the
of
H
The permutation character
100 ~
cosets
correspond-
ing to the permutation representation can be found from the character table.
x=
~O
+ '1 + ' 7 ·
is of rank
It is
The permutation representation
3, and as a 7-element fixes exactly
two points, it readily follows that the stabilizer of a point,
G ,has orbit lengths a
1, 36 and 63 .
The permutation representation of orbit of length
36
H
on the
is on the cosets of a subgroup
FINITE GROUPS
82 of order
168.
Such a subgroup can be found and
is unique to within conjugation in the permutation character of its character on the
63
H
Knowing
on the
36 orbit,
orbit is determined,
and it follows that the tion of
H
H.
63 orbit is the representa-
on cosets of the centralizer of an
involution. It is now only necessary to show that this representation can be extended uniquely to a group of order
604,800 .
In
H
the normalizer
of a 7-Sylow group is of order (a, c>
(1)
a
7
and
H S
::::
d
:::: 1, c
-1
2 3 a , c :::: 1 .
ac
a, c
and two further elements
are given at the end of the paper.
(a, b>, d
2
::::
1, dcd
of the two letters
a group of order In
and
where
Permutations for b
21
G
::::
00
-1
and
and the stabilizer 01
is
S
::::
(a, d> ,
168
the normalizer
group is of order
C
Here
42
,
N (7) G
and so
G
of a 7-Sylow must contain an
83 involution
t
such that a
tat
( 2)
Furthermore
-1
G
=
tc
ct .
= , as
H
must
be maximal. It remains to show that unique and that the group 604,800.
t
exists and is
has order
The latter was done with the help of
Peter Swinnerton-Dyer on the Titan computer at Cambridge University.
The simplicity of
G
follows from an easy argument using the simplicity of
H.
From the character table, 100
letters.
letters tc
= ct
by
c.
Hence
(00),
t
(01)
fixed by
tat
=
a
of the values in 2 4
a
the two further letters As
-1
2
tb a tb t determined.
fixes
(00)
t
must move all
must interchange the two and also, as (12,
fixed
(31)
this determines
the 7-cycles of a containing 16
t
(12) and (31)
t
on With
determined, we find that
and has
7
further values
This is sufficient to determine the
84
FINITE GROUPS
permutation completely as an element of
H
and in
fact 363 a b ada .
(3) ?elations ~:~e
t~at
t
( 2)
and
(3) are now enough to deter-
completely.
The computer verified
the subgroup of
exactly g~oup
H
and so
of order
G
=
fixing
604,800
00
is
is the desired
and is unique.
Suzuki has also constructed the group as a subgroup of index
2
a certain graph with ~as
over
of the automorphism group of 100
points.
Jacques Tits
a similar construction using certain geometries GF(4)
Recently John Lindsey has found the
matrices for a 6-dimensional projective representation of
G
over the complex numbers.
85
element
e
a
J
R
K
TJ
TR
order
1
7
2
4
8
6
12
C (X)
g
7 1920 96
8
24
12
~O
1
1
1
1
1
1
1
~1
36
1
4
4
0
1
1
~2
90
-1
la
-2
0
1
1
~3
160
-1
0
0
0
0
0
~4
225
1 -15
-3
-1
0
0
~5
288
1
0
0
0
0
0
~6
300
-1 -20
4
0
1
1
~7
63
0
15
3
1
0
0
~8
126
0
14
2
0
-1
-1
~9
70
0 -la
2
0
-1
-1
~10
70
0 -la
2
0
-1
-1
~ll
175
0
15
-1
-1
3
-1
~12
224
0
0
0
0
0
0
~13
224
0
0
0
0
0
0
~14
14
0
-2
2
0
1
-1
~15
14
0
-2
2
0
1
-1
~16
21
0
5
1
-1
-1
1
~17
21
0
5
1
-1
-1
1
336
0
16
0
0
-2
0
~19
189
0
-3
-3
1
0
0
~20
189
0
-3
-3
1
0
0
*18
FINITE GROUPS
86
element
TIJ
order
10
C(X)
20
1
1
1
1
1
1
1
-1
-1
o
o
o
o
,2
0
0
6
o
1
1
,
0
0
4
1
-1
-1
0
0
5
-1
o
o
0
0
4
1
-1
-1
0
0
o
o
o
o
0
0
-1
-1
-1
-1
-1
-1
6
o
1
1
-2
1
'" 0 '1
~ ~
.et '5 'il 'f
6 7
~8
'il
9
L o
0
-2
1
0
0
-5
1
o
o
~12
0
0
-4
-1
1
1
Vu
0
0
-4
-1
1
1
'+'14
6
2
-1
-6
V1S
6
2
-1
-6
'+'16
0
0
-3
0
-6
'+ 17
0
0
-3
0
-6
'18
o
o
'19
o
'20
o
'+ 10
Vl l
2 1
6 6
1 2
1 2 2 1
-6
-6
2 1
-6 -6
o
1 2
87 TI
TI
3
5
5
36
300
element
T
T
order
3 1080
C(X)
1
2
TI
rr
1
2 1
,...2
TIT
.. T
5
5
15
15
300
50
50
15
15
~O
1
1
1
1
1
1
1
1
~1
9
0
-4
-5
1
1
-1
-1
~2
9
0
5
5
0
0
-1
-1
~3
16
1
-5
-5
0
0
1
1
~4
0
3
0
0
0
0
0
0
~5
0
-3
3
3
-2
-2
0
0
~6
-15
0
0
0
0
0
0
0
~7
0
3
3
3
-2
-2
0
0
~
9
0
1
1
1
1
1
1
~9
7
1
58
0
0
-8
~10
7
1
58
0
0
-8
'~11
-5
1
o
o
o
o
o
~12
8
-1
1-48
8
8
~13
8
-1
1-48
~14
5
-1
38
~15
5
-1
38
~16
3
0
3+ 8
~17
3
0
3+8
~18
-6
0
-4
~19
0
0
38
'1J
0
0
38
8
20
58
1
58
2
2
1
o 2 1
1-48 1-48 38
1
38
2 1 2
1 2
1+8
1
3+8
-4
-28 1+8
2
3+8
-28
2 1
28 28
1 2 1 2 1 2
1
2 1
1+8 1+8
-28 -26 1+8 1+6 26 26
2 1 2 1 2 1
1
1 2
1+6
1+e
2
1
1 2
1
8
2
-8
8
0
0
0
0
8
1
8
2
2
-8
1
2 1
8 8
2 1
-1
-1
o
o
o
o
FINITE GROUPS
88
Permutations a =
(00) (01) (02,03,04,05,06,07,08) (09,10,11,12,13,14,15) (16,17,18,19,20,21,22) (23,24,25,26,27,28,29) (30,31,32,33,34,35,36) (37,38,39,40,41,42,43) (44,45,46,47,48,49,50) (51,52,53,54,55,56,57) (58,59,60,61,62,63,64) (65,66,67,68,69,70,71) (72,73,74,75,76,77,78) (79,80,81,82,83,84,85) (86, 87, 88, 89, 9 0, 91, 92) (93, 94, 95, 96, 97, 98, 99)
b
(00) (01,02,09,16,23,20,30,17) (03,35,13,29,31,24,25,11) (04,26,27,07) (05,21,14,10) (06,19,32,36) (08,18,28,22,15,12,33,34) (37) (38,92,67,77,99,89,49,84) (39,59,82,46,88,54,52,68) (40,55,47,81,75,95,61,78) (41,44,63,58,72,65,50,51) (42,76,53,93,86,80,79,57) (43,85,48,70,96,83,66,94) (45,97) (56,87) (60,71,91,69,90,73,64,74) (62,98)
c = (00) (01) (02,28,21) (03,23,18) (04,25,22) (05,27,19) (06,29,16) (07,24,20)(08,26,17) (09,13,14) (10,15,11) (12) (30,36,34) (31) (32,33,35) (37,50,63) (38,45,60) (39,47,64) (40,49,61) (41,44,58) (42,46,62) ( 43, 48 , 59) (51, 65 , 72) (52, 67, 76) (53, 69, 73) (54,71,77) (55,66,74) (56,68,78) (57,70,75) (79,89,96) (80,91,93) (81,86,97) (82,88,94) (83,90,98) (84,92,95) (85,87,99)
89
d
=
(00) (01) (02) (03,05) (04,25) (06) (07,20) (08,17) (16,29) (18,27) (19,23) (21,28) (22) (24) (26) (09,14) (10) (11,15) (12) (13) (30) (31) (32,35) (33) (34,36) (37,44) (38,88) (39,64) (40,90) (41,50) (42,95) (43,97) (45,82) (46,92) (47) (48,86) (49,83) (79,89) (58,63) (59,81) (60,94) (61,98) (62,84) (80,85) (87,93) (91,99) (96) (51,72) (52) (53,68) (54,71) (55,66) (56,69) (57) (65) (67,76) (70,75) (73,78) (74) (77)
t
=
(00,01) (02,74) (03,73) (04,72) (05,78) (06,77) ( 07 , 7 6) (08, 7 5) (0 9 , 34) (1 0 , 33) (11, 3 2) (1 2 , 31 ) (13,30) (14,36) (15,35) (16,71) (17,70) (18,69) (19,68) (20,67) (21,66) (22,65) (23,53) (24,52) (25,51) (26,57) (27,56) (28,55) (29,54) (37,91) (38,90) (39,89) (40,88) (41,87) (42,86) (43,92) (44,99) (45,98) (46,97) (47,96) (48,95) (49,94) (50,93) (58,85) (59,84) (60,83) (61,82) (62,81) (63,80) (64,79)
90
FINITE GROUPS
Bibliography :lj
Hall, M. Jr.:
A search for simple groups of
orders less than one million.
:2.
Hall, M. Jr. and D. Wales: of order
3.
604,800.
To appear. The simple group
To appear.
Janko, z.:
Some new simple groups of finite
order, I .
To appear.
LE GROUPE DE JANKO D'ORDRE 604.800. J. Tits
Soit groupe
H
l'hexagone generalise associe au
G (2) 2
points de
H
On sait [3] que l'ensemble des
points d'une hyperquadrique sur
E2.
Un plan de
Q
contient aucune droite de nocmaux distincts toute droite de une droite de est associe
a
Soient
a
peut etre identifie
p, p' Q
H
Q
l'ensemble des
de dimension
est dit normal s'il ne H; deux plans sont dits associes si
rencontrant
p
et
p'
A
un et un seul autre plan normal. H'
=
est
On montre que tout plan normal
l'ensemble des droites de
l'ensemble des paires de plans associes, point et
5
H' UP U {oo}.
la relation binaire symetrique
91
Dans
00
H, P un
A, considerons
I C A
x
A
definie
?2
FINITE GROUPS
20mme suit:
pour
x)
( 00,
I
E
(d, d')
E
d, d' E H'
I
~
X
E
et
P
les droites
d
et
d'
sont
disjointes et non opposees dans
(i.e. il existe une
H
et une seule droite de rencontrant ~
Soit
I.
elements de
J
automorphismes (resp.
(resp. de
n
p.1
et
K'
Soit
intersections
Pi
une droite de
Q.
K
d'
);
'1.0
le groupe des permutations de
J
conservant
J'
d
d
H
n
est
qj
A
le groupe des
induits de fa90n evidente par les ("Collineations") de
K'
H
et soit
le groupe des elements de
J
) qui SOnt des permutations paires
A.
THEOREME.
On a
(J:J' )
(K: K' )
2.
Le groupe
93
J'
est un groupe transitif de permutations de
l'ensemble d'ordre J
a
(resp.
J'
elements
604.800.
A
i
il est simple,
Le sous-groupe des elements de
laissant fixe le point
J'
sous-groupe de
100
K
(resp.
K
I
est le
00
Tout automorphisme
).
s'etend de maniere unique en un automor-
phisme de
J , et tout automorphisme de
interieur, de sorte que
~
Aut (J')
J
Jest
.
Un etape essentielle de la demonstration consiste evidemment
a
montrer l'existence d'une
permutation impaire de
A, appartenant
ne conservant pas
Elle ressort du
00.
LEMME.
a
J
et
11 existe une
permutation involutive
a
proprietes suivantes, ou
de K
A
possedant les
= (ql' q2)
E
P
et
d E H'
(i)
a (00)
=
a(TI)
TI
00
(ii)
si
( TI ,
K)
E
I
,
alors
( iii)
si
( TI ,
K)
~
I
,
alors, apres permu-
a (K)
=
K
tation eventuelle de ql
et
q2
, les
94
FINITE GROUPS intersections
Pi
n qi
sont des points, joints par une droite et (iv)
si
(TI,
d)
a(K)
E l , a(d)
=
d'
est l'unique paire
d
(v)
si
(TI,
d)
~
I
d' E H',
n
p.
1
;
, il existe une seule droi te
dIE
rencontrant d , et
a(d)
H'
Pl' P2
et
est
l'unique droite
E HI
distincte de
et
d
d',
et contenant le point d
L'involution appartient
a
n
d'
a , qui est manifestement unique, J
Les demonstrations sont assez longues, parce G~e ~ais
requerant de nombreuses distinctions de cas, elementaires. Les resultats precedents ont ete suggeres par
95
la lecture de [1].
La representation, decrite
ici, du groupe de Janko comme groupe de permutations de degre
100
est bien entendu equivalente
a celle obtenue par M. Hall au moyen d'un ordinateur; la connaissance de celle-ci a aussi oriente nOs recherches.
REFERENCES 1.
D. Higman and C. Sims, A simple group of order
2.
44.352.000.
To appear.
Z. Janko, Some new simple groups of finite order, I.
To appear in the Proceedings of a
Conference on Group Theory (Rome, 1967). 3.
J. Tits, Sur la trialite et certains groupes qui s'en deduisent, Publ. Math. I.H.E.S., 2
(1959), 13-60.
Bonn, Decembre 1967
LINEAR GROUPS OF DEGREE
6
AND THE
HALL-JANKO GROUP John H. Lindsey 11
Finite groups
G
having a faithful,
irreducible, quasi-primitive, unimodular representation of degree
6, over the complex field, have
essentially been classified. extension by
is a central
AS' A , A , L (7), L 2 (11), L 2 (13), 6 7 2
L (4), U (3), U (3), 0 (3), 3 4 3 5 group of order
G
or the Hall-Janko
604,800, or
over one of these groups.
G
is of index
2
The only questions left
open are the automorphism groups of the central extensions by
A 6
and
L (4) 3
of the central extension of representation of degree
and the uniqueness 2
6
by
L (4) 3
with a
6
In the course of classifying groups of degree
6, the following generalization of Feit's
theorem was proved: 97
98
FINITE GROUPS
Let h = g
TI
G
,
be a group of order
~
TI
{primes
> r
abelian sUbgroup of
G
- l}
2
either A.
or
B.
, and
of order
X
a faithful representation not a power of
glh
h.
g H
where is an
Let
of degree
G
have
r - 1
over the complex numbers.
Then
G
has a normal subgroup of order
h
or
P
hip,
a prime number,
There exists a subgroup of index
1
or
2
with
GO
of
G
GO
isomorphic to a central extension by
L 2 ( r)
A group of index of
Z6
by
U (3) 4
matrices in
(1)
2
~(w»
over a central extension
where
w
is a primitive
1.
All diagonal matrices of order
1 ,
determinant ( 2)
L 2 (r)
was generated by the following
GL(6,
third root of
x
All permutation matrices, and
(3 )
1 w - w
1
1
1
1
w
w
1
w
w
0
0 -1
-1
-1
-1 -1
-w
-w
-w
-w
3
and
99
It is shown that these generate a finite group by examining the
126
conjugates of a
permutation matrix corresponding to a transposition. 45
conjugates are monomial.
conjugate to order
16 -
3 , where
entries are
(1/3)M M
81
by diagonal matrices of
is a matrix all of whose
1
There is a unique group of order
conjugates are
2
and
Gl/Z
G l
with center
Z
G , the Hall-Janko group
~
satisfying the following: (A)
The Sylow 7-subgroup of
G
normalized by an element of order (B)
is
.
4
The inverse image, under the homo-
morphism
G l
of order
6048
->-
of a subgroup
Gl/Z
is isomorphic to
of
U (3) 3 Z
x
G
U (3) 3
G l
has two conjugate, faithful, irreducible representations of degree
6
in the complex field.
representations can be written in character table of
Gl
~(15,
can be given.
These
17 ).
The degrees
of the faithful irreducible representations of are
The
G l
6, 6, 64, 64, 50, 50, 216, 14, 84, 126, 126,
252, 56, 56, 448, 350 Existence of (mod 3)
G l
and
336.
was verified by taking
a representation of degree
6.
This
FINITE GROUPS
100 ~odular ~as
a
~hen,
representation restricted to 3
Z x U (3) 3
dimensional invariant subspace,
under
G l
generators of
, G
l
V
has
100
V.
images, which the
permute exactly as their images
permute the letters described in the Hall-Wales paper. Unique unitary matrices over the complex :ield were obtained for a
6
dimensional
representation of generators of ~ormalizer ~ormal
G
l
after the
of a Sylow 7-subgroup was written in a
form.
ON THE ISOMORPHISM OF TWO GROUPS OF ORDER
44,352,000
Charles C. Sims
1.
INTRODUCTION At the Group Theory Symposium held in Urbana
November 24, 1967 G. Higman described a simple doubly transitive permutation group of degree
176
with the same order and character table as the simple group discovered by D. G. Higman and the author.
In this note a proof of the isomorphism
of these two groups will be sketched.
The proof is
computational in the extreme and was carried out entirely on a computer.
2.
THE GROUP OF G. HIGMAN G. Higman defined his group as the auto-
morphism group of a "geometry" whose objects are 101
102
FINITE GROUPS
points and subsets of points called quadrics such that (i)
There are
(ii)
176
points and
Each quadric contains
each point is in (iii)
50
176
50
quadrics.
points and
quadrics.
Any two distinct points are in
14
quadrics and any two distinct quadrics have
14
points in common. (iv)
There is a polarity interchanging points
and guadrics, preserving incidence. Higman's definition of the geometry is equivalent to the following:
Let
A subgroup of the symmetric group be called a
PGL(2, 5)
orbits of length C
of
168
1, 1
B
{1,2, ... ,8}
=
S8
on
if it has order and
PGL(2, 5) 's
6
S8
B 120
will and
has one class
The set of points in
the geometry is P
~he
B U C •
quadrics are in
~oints.
If
B U {H I~
H
€
1-1
correspondence with the
b
€
B , then the quadric
I
H
€
C , then
C, H
b
H
=
L(b)
is
b}
has orbits on
C
of length
103 1, 5, 12, 30,
60
and
The quadric
60
contains the fixed points of orbits of length Let
Q
1, 5, 12
H
on
B
and
30
of
denote the set of quadrics.
morphism group of the geometry of order
44,352,000.
is isomorphic to PSU(3, 52)
L(H) and the H
on
C.
The auto-
(P, Q)
is simple
The stabilizer of a point
P2:U(3, 52)
, the extension of
by a field automorphism.
Higman distinguished certain other subsets of points called conics. ~ich
points are
1100
These are sets of
are contained in
conics.
B
8
quadrics.
8
There
is a conic and the only
automorphism of the geometry fixing a conic point-wise is the identity.
3.
THE GROUP OF HIGMAN-SIMS Let
Sand
B
denote the sets of points
and blocks, respectively, in a fixed Steiner system S(3, 6, 22) (i)
(ii) of
Thus
Is[:= 22. B
is a set of
77
6-element subsets
S . (iii)
Any three distinct points are contained
104
FINITE GROUPS
in a unique block. Define a graph
*
where
~
with vertex set
is a new symbol.
points in
Connect
*
to the
S , connect each point to the blocks
containing it and connect two blocks if they are disjoint.
The automorphism group of
~
88,704,000
and contains a simple group
index
The stabilizer of a point in
2.
isomorphic to
has order G
of
G
is
M . 22
THE ISOMORPHISM
4.
Let us denote the group defined in section 2 by
GH
and the group defined in section 3 by
HS
To prove that these two groups are isomorphic it is enough to show that isomorphic to
M 22
character table of
GH
contains a subgroup
By inspection of the M 22
possible degree less than
we see that the only 100
permutation representations of 56
and
of an
77
Thus representing
of transitive M 22 GH
are
1, 22,
on the cosets
M , we must get a primitive rank 22
3
group
105 of degree
100
with subdegrees
1, 22
and
77.
It is not hard to show that this group would have to be an automorphism group of the graph defined in section 3. To show that
GH
contains an
construct a geometry on which
M 22
M 22
we
acts and prove
that this geometry is isomorphic to the geometry defined in section 2.
M 22
subgroups isomorphic to
has two classes of
A
There are
7
176
subgroups in each class and the two classes are interchanged by an outer automorphism of
M 22
A sUbgroup in one class has orbits of length 35
and
126
on the other class.
15,
We define a
new geometry in which the points are the sUbgroups in one class of
A 's 7
and the quadrics are the
subgroups in the other class. quadric
K
A point
and a
H
are incident if
IH n KI
72
or
168.
We must now show that the geometry
(P, Q)
of section 2 is isomorphic to the geometry (pI,
Q')
just constructed.
that we turn to the computer.
It is at this point The two geometries
were explicitly determined and a particular
106
FINITE GROUPS
isomorphism exhibited.
The construction of the
geometries is relatively easy and it is sufficient to say that programs were written to compute the two
176
176
x
incidence matrices of the
geometries. Determining an isomorphism was somewhat more difficult.
G. Higman showed that the
point-wise stabilizer of a conic
B
in
(P, Q)
is trivial by showing that there is a "natural" identification of the points in set of
PGL(2, 5) 's
P - B
with the
in the symmetric group on
This identification is obtained as follows: i,j
E
B,
i
x
in
P - B
8
quadrics.
t
j
,
let
such that
C..
1.J
{i,
k t that
i, j
{i, y, z}
of order
j, x}
is contained in
y
(i,
z
in
C
such
jk 3
Given an element
5
in the symmetric group on
(i, b r )
(i, b ), l
j)
in
is contained in more than
composition of the maps and
be the set of points
, with the unique point
quadrics.
For
Also, define a map denoted by
which interchanges the point
B
B , the
(j, b ),
defines a permutation
r
f
r
on
107 The product
has a unique fixed point with the unique
x
group on g
B
x
PGL(2, 5)
which fixes
c l..]
on
i
Identify
in the symmetric and
j
and contains
•
A computer program was written to look for conics and to construct this identification.
The
only input to the program was the incidence matrix of
(P, Q)
The program was then given the
incidence matrix of
(P I,
Q')
instead.
If the
geometries were isomorphic, then a conic would be found and the identification determined.
If the
geometries were non-isomorphic, there were several possible results: (1)
No conic would be found.
(2)
A conic
identification of PGL(2, 5) 's
B'
would be found but the
P' - B '
with the set of
in the symmetric group on
G'
could
not be carried out. (3)
A conic
B'
and the identification
would be found. As it turned out the computer found a conic
108
FINITE GROUPS and constructed an identification of
3'
·.·:i th the set of
group on in
PGL (2, 5)' s
B'
(P, Q)
~
A map was chosen.
of the synunetric groups On
B'
onto a conic
pI
verified that
onto ~
P.
B'
B
induced an isomorphism B'
and
identifications, one could extend a map of
-
in the synunetric
of ~
pI
using the
B ~
uniquely to
A separate program
took quadrics to quadrics and
thus was an isomorphism.
5.
PW(3,
52) 'S
IN
HS
We close with a construction of the graph in section 3 which shows that subgroups isomorphic to has three classes of
HS
contains
PZU(3, 52)
A ' s. 7
PSU(3, 52)
Take the vertices of
the graph to be the subgroups in two of these classes. (a) H
n
KI = (b)
H
n
KI =
Connect two vertices Hand
Hand
K
K
are conjugate and
K
are not conjugate and
360
Hand 168
Rutgers University
if
A SIMPLE GROUP OF ORDER
898,128,000
Jack McLaughlin
Let
U
denote the unimodular group of a
polarity of unitary type on projective space of dimension group
3
over the field of
U4 (3)
9
elements (the ~
in Artin's notation), and let
denote the class of totally singular line stabilizers in
U.
primitive, of rank
The action of
3 , with subdegrees
If we ask for a primitive rank which a point stabilizer is orbits for
U
is
on Y
U
U
is
1, 30, 81.
3
groups in
and one of the
Y, the conditions worked out
by Donald Higman in [2] tell us that
162
is a
possible value for the other non-trivial orbit length. E. M. Hartley in [1] showed that
U
contains
the unimodular group on the projective plane of 109
110
FINITE GROUPS
order
4
(the group
index
162).
Let
isomorphic to
L (4) ) as a subgroup (of 3 be a class of subgroups
Y
L (4) 3
in
U.
One can verify the
following: (1) and
U
has
orbits on Y
2
x
Y , say
0'
0" . The action of
(2) or rank
U
on
3 , with subdegrees
Y
is primitive,
1, 56, 105 .
Following Higman-Sims [3] we make a graph
* U YuY.
whose vertex set is member of
Y.
Join
S
E:
Y
of the S-orbit of length L
E:
Y
with
(S, L)
E:
0'
to
30 .
On Join
members of the L-orbit of length to the
S
E:
Y
with
(S, L)
E:
*
Join
to each
* , the members Y, and the L 56
0'
has a transitive automorphism group. of the title is a subgroup of index
E:
Y on
to the
Y, and
The graph
!§
The group 2.
REFERENCES 1.
Hartley, E. M., Two maximal subgroups of a collineation group in five dimensions, Proc. Carob. phil. Soc. 46 (1950), 555-569.
111
2.
Higman, Dona1d G., Finite permutation groups of rank
3.
3 , Math. Z. 86 (1964), 145-156.
Higman, Dona1d G. and Charles C. Sims, A simple group of order
44,352,000,
Math. Z.
105(1968) 110-113.
The University of Michigan
A SIMPLE GROUP OF ORDER
448,345,497,600
Michio Suzuki
We construct a simple group
S
of order
448,345,497,600 . Comparison of order shows that
S
is a new simple
group not found in the list of simple groups published so far.
What we have proved is the
following theorem.
THEOREM 1. extension E 2 (4)
There exists a primitive transitive S
of degree
1782
of the simple group
, the Chevalley group of type
G 2
over the
field of four elements. Since
E (4) 2
is simple and there is no
characteristically simple group of order 113
1782,
FINITE GROUPS
114 Theorem 1 yields
COROLLARY.
S
is a simple group.
Thus the group group on rank
3
1782
S
is defined as a permutation
elements and as such
S
is of
in the sense of D. G. Higman [3].
The
parameters are k
416 ,
1365,
in the notation of [3].
A
100
and
~
The permutation group
defines a strongly regular graph
r
with
96
S
1782
vertices, such that the group of automorphisms of 1
contains
S
The proof of Theorem 1 depends on
the construction of a graph which turns out to be
r The construction of the graph depends On various properties of the simple group
E (4) 2
Among them the following result is crucial.
THEOREM 2.
The simple group
E (4) 2
has a
primitive permutation representation of degree such that it is of rank
3
416
and the stabilizer of
a point is isomorphic to the simple group of order
115 604,800
recently constructed by M. Hall [2].
Let
6
be the graph associated with the
permutation representation of
=
Let
E (4) 2
be the set of sUbgroups of
in Theorem 2. E 2 (4)
which
are conjugate to the center of a Sylow 2-group of E (4) 2
r
We construct a graph
r
vertices of we denote
as follows.
The
are a distinguished point, which
(00)
, vertices of
6
and elements of
=: r
(00) U 6 U
r
The edges of (a) 6
The point but none of
(b)
r
=
are defined as follows: (00)
is joined to every point of
=;
Two vertices of
6
are joined by an edge in
if and only if they are joined by an edge of
(c)
If
a s 6
and
b s
=,
a
and
b
are joined
by an edge if and only if the stabilizer of E 2 (4)
a
in
contains a non-identity element of the
subgroup
b
(d)
u
If
we join [u, v]
L
~
and
v v
are two subgroups of
u
and
1
but there is a subgroup
=,
then
by an edge if and only if w
of
=
such
116
FINITE GROUPS
that
=
(u, w]
=
(v, w]
1.
Here
the commutator subgroup of It is proved that
u
and
Aut f
denotes
(u, v]
v.
is transitive on
the set of vertices and contains the simple group S
of index
2
which satisfies the condition of
Theorem 1. The proof of Theorem 2 is in turn reduced to a similar proposition about the Hall-Janko group. The graph
6
is constructed from the representation
of the Hall-Janko group of degree
100
In fact
we may begin the construction starting from the trivial graph
f
consisting of four vertices and
l
no connecting edge.
The group
the symmetric group
S4
from
6
f
G 2
14
=
before.
vertices.
Aut f
2
G 2
construction the group It is the stabilizer of on G l closure of
Ll
Let
We construct
S4
Since
in
9 ,
It is not hard to verify
~
PGL(2, 7)
G2
By
is a subgroup of
G l ( co)
G 2
in the natural action
We may define
f
2 Ll
is
l
is transitive on the set of
vertices and in fact
of
f
in a similar way as we constructed
L
and
contains that
2
= Aut
on four letters.
be the set of involutions of the graph
G l
The set
as the normal
L2 L2
consists
f
117 of
21
involutions of
define
f
, G , =3 3
3
PSL(2, 7)
similarly.
We can
Continuing this
process we obtain 36
If 4 1
100
Ifsl
Here
,
If 3 1
HJ
=
=
G 3 G 4
,
416
G S
E 2 (2) Aut HJ ;
=
Aut E (4) 2
denotes the simple group of Hall-Janko.
The graph
f
is the graph
S
6
defined just after
Theorem 2. The proof of a crucial point, the group
G.
acting transitively on the set of vertices of
f.
1
1
depends on various properties, among which we single out two. Let
x
be a vertex of
stabilizer of to L 1- 1
G i l
x
in
G.
f.
1
Then
1
involutions of outside
H
and this isomorphism sends Thus, in the embedding of
.
and
=.1- 1
H
be the
is isomorphic =.
G _ i l
1
n H onto into
G.
1
do not fuse to involutions
=.1- 1 .
The second, more important, property is the following. of
x.
Let
As before let i
x
L (7) 2
, and
If (w) I
Remark.
with
w , and five classes of
elements of order (j
S
3.7 3
The proof leading to the above results
will be published in a forthcoming paper.
In this
FINITE GROUPS
:1.24 ~aper,
the existence problem will be attacked.
particular, the character table of
G
will be
computed. In
t~e
proof of the above theorem a new
group-orcer formula of J. G. Thompson plays a crucia::' :-::le.
In
AN ANALYSIS OF GROUP REPRESENTATIONS S. B. Conlon
Let
G
be a finite group and
tative ring with identity.
Let
Jt
R
a commu-
denote the
category of finitely generated left RG-modules. The representation ring is formed from integral
a (RG) (~
of the isomorphism classes subject to the relations ever
M "" M' Gl M"
(algebra
A (RG) )
-linear) combinations {M}
{M}
for
= {M'}
M
in
.1,
+ {M"}
when-
and in which mUltiplication is
given by the "tensor product representation," i.e. {M}·{M'} ~heorem
{M ~ M' }
holds in
If the Krull-Schmidt
Jt, then we have a natural
imbedding of the isomorphism classes in
a(RG)
and of
a(RG)
in
of Jt
{M}
A(RG)
In the classical case with
R
~
we know
that character values separate not only the 125
126
FINITE GROUPS
isomorphism classes of
~,
but, if extended
linearly, also separate the elements of
A(RG)
One can ask the same question for general When
R
is a complete
~_ocal
the Krull-schmidt theorem holds in
R.
noetherian ring ~
and Green's
work on vertices and sources and transfer goes through.
In [1]
it is shown that, provided one
has an adequate knowledge of the possible (absolutely indecomposable) sources that can arise, the elements of
A(RG)
can be systematically
separated and the question is really one of the sources. The reduction involves two ideas.
First, a
close look is taken at the decomposition of an G L
induced module source of vertex
RG 0
RP
L ,where
L
is a
P , by means of rings of endo-
morphisms and the notions of ideals and radicals in categories.
The analysis elucidates Green's
transfer theorem. The second idea is that of a canonical decomposition
A(RG)
into two-sided ideals
~
Ap(RG)
Ap(RG)
, of
, where
~
complete set of non-conjugate p-subgroups G
and where
p
A(RG) is a P
is the characteristic of the
of
127
residue field of is easier.)
R.
(The characteristic
The ideal
A" (RG) P
0
has a basis corres-
ponding to the different indecomposables in vertex of
P
case
At of
Actually the projection of an element
A(RG)
onto the linear subspace of
A" (RG) P
corresponding to a given source is determined in [1].
However it is the ideal (or multiplicative)
structure of
A(RG)
which enables the analysis to
go through. However this ideal decomposition is only a particular instance of a much more widely valid (R
arbitrary) and finer decomposition of
flowing out of the coset structure of If
H < G , and if
RH-module, then
IH
G
IH
A(RG)
G.
is the trivial
only depends upon the
conjugacy class of the subgroup
H
in
G
By
the Mackey formula for the tensor product of induced modules we have that
If
H
is conjugate to
Consider the xH .
~
H'
in
-vector space
G B
write on the symbols
We define a multiplication in
B
by
128
FINITE GROUPS
Then
B
is a
-algebra and is called the
~
Burnside algebra.
It is discussed in [2] and [3].
It is semisimple and is the ring direct sum of copies of
~
idempotent
, each ideal generated by an IH
of subgroups ~
x
H
associated to each conjugacy class
H
of
-algebra homomorphism
We have a natural 8 : B
~
~ lH G , with the identity of
the identity of A(RG) 1 :
G.
L
A(RG)
A(RG) B
given by
mapping onto
Thus the identity
1
of
is written as the sum of idempotents 8(I ) H
and we have corresponding ideal
decompositions of
A(RG)
In the particular case when
R
is a complete
local noetherian ring with residue field of characteristic
p,
p'-cyclic extension of a p-subgroup Then the idempotent generator of by the sum of those
8(I H)
G
of
A"p (RG)
where
a complete set of non-conjugate (in of
P
H
G is given
runs through G)
subgroups
which are p'-cyclic extensions of the
p-subgroup
P .
129 REFERENCES 1.
Conlon, S. B., Relative components of representations, to appear in the J. of Algebra.
2.
, Decompositions induced from the Burnside Algebra, to appear in the J. of Algebra.
3.
Solomon, Louis, The Burnside algebra of a finite group, J. of Combinatorial Theory, 2 (1967), 603-615.
CENTRALIZER RINGS AND CHARACTERS OF REPRESENTATIONS OF FINITE GROUPS C. W. Curtis and T. V. Fossum
Throughout this paper finite group, KG
and
KH
H
G
will denote a
an arbitrary subgroup of
G
will be the group algebras of
G
and
H
respectively, over an algebraic number field
K
which is a splitting field for both
KH
KG
and
The first main result (Theorem B) gives some
orthogonality relations in the centralizer ring of a representation of representation of
G H
induced from a linear This result is applied to
give a new arithmetical result (Theorem C) on the degrees of irreducible characters of Let
~
G
1
be an irreducible character of
H.
1. The result on the degrees of characters was obtained independently, and by a different method, by Gordon Keller. 131
132
FINITE GROUPS
Then
W
where
e
is afforded by the left
KH-module
is a primitive idempotent in
centralizer ring
E = eKGe
an algebra with identity simple subalgebra of
~
HO~G
KHe ,
KH.
The
(KGe, KGe)
is
e , which is a semi-
KG
Representations,
characters, and degrees of characters are defined for
E
as well as for
KG
irreducible character of
is the multiplicity of
The character
LEMMA.
character of restriction character
E /;'E
to G
E
/;,
in
ep
wG
of
is an
.
is an irreducible
E
ep
is the
of a unique irreducible
such that
We remark that for
/;,
KG ,
i f and only if
of
/;,
If
( /;,
,
wG )
> 0
.
S-rings, this theorem is
due to Tamaschke [6]. The following notations will be used in the remainder of this paper: x H
=
x
-1
HX,
for
x
E
G ,
133
[H
ind x
We shall also assume that of
is a linear
~
H ; then the primitive idempotent
that
KHe
affords
THEOREM A.
be a linear character of
~
KH-module
= U.lE I
KHe
by (*)
Let
distinct
(H,H)-double cosets
X
E
G
O.l
,~
O.
J
x
=
H
e
x
H
H ,
is given are the
O. l
in
G
Let
such that for some
j
on
~
where
, where the
be the set of indices
I
suc~
can be given explicitly by
~
Let
afforded by the
C
KH
E
\ 'I>(h-l)h. = I Hj-l LhEH '¥
e
J
e
characte~
This condition
depends only on the double cosets and not on their representatives. and let
=
a.
J
For each
j
(ind x.) ex . e J
{a. J
j
E
J}
form a basis for
{D. J
j
E
J}
is invariant under
=
b.
J
(ind x .) ex . J
J
-1
e.
{b.
E
: j
J
J
and
{(ind x.)-lb.} J
respect to the form
J
(x, y)
EO.
J
J
The set
eKGe . 0 ~ 0- 1 E J}
The algebra
Frobenius algebra and, letting {a .}
X.
The elements
J
there is a second basis
pick
E J
xl
E
=
so that
where is a 1 , the bases
are dual bases with
=
Sl
I
where
FINITE GROUPS
134
then the constants of structure algebraic integers in
are all
K.
The computations in the proof are similar to the arguments in Section 1 of [3] and will be omitted.
THEOREM B. ~
Let
be an irreducible character of
(~,
that
Assume the notations of Theorem A.
> 0
1jJG)
G
such
Then
[G ·.H]-l"'(l) S
\' L'
JE
('In d x, )-l"'(b) a.
J
J
L,
•
J
J
where central primitive idempotent in to
~
of
G
If
~
and
~'
both appearing in
KG
corresponding
are irreducible characters 1jJG
with positive
mUltiplicity, then the following orthogonality relations hold:
~
~'
The proof of the orthogonality relations follows from the observations that
135 I:;(E(I:;)e) I:;
t-
I
I:;
( 1:;,
1jJ
G
and
)
I:;
I
o
(E (I:;) e)
. In case
VG ) =
(1:;,
1 , the element
Theorem B is a primitive idempotent in that
KGE(I:;)e
THEOREM C. Let
I:;
that
if
affords
I:;
(by Janusz
KG
in
such
[5]).
Assume the notations of Theorem A.
be an irreducible character of i;;(e)
E(I:;)e
=
I:; (1)
\jJ
(1:;,
G
)
> 0
[G:H]~cm.
JE
J
C(l)
(ind x,) J of
{ind x,} J
such
Then
.
\ [G:H]c(e){L' J JE
Moreover the degree
G
C
-1
I:;(b·)c(a.)} J J
-1
divides
.
The first statement follows from Theorem B. For the second statement, suppose valuation on
K
p
rational prime, and let
R
in
v
corresponding to
principal ideal domain.
=
is a
extending the p-adic valuation On
the rational field, where
K
v
is an arbitrary be the valuation ring Then
R
is a
By Theorem A,
Ra, is an R-order in E containing the J set {b. : j E J} is One can show that I:; (b . ) J J in R for each j E J Let
E'
LjEJ
136
FINITE GROUPS 9, cm .
{ind x.}
]EJ
. L]EJ
w == Then
w
E
But then
p
Q
E'
R
9, I;, (b . ) a . ]]
, and by Theorem B,
It follows that
, since
0:
-1
]
E
for
where
(ind x.)
is in
central in E
and write
]
0.
I;,
w
(w) ==
is ( I;, ,
wG)o:
is a splitting field for
K
== I;, (1) -l(G:H] 9,
is in
is the rational field.
was arbitrary we conclude that
Q
n R
Since the prime 1;,(1)
divides
(G:H]9, Remarks.
Ito's
2
Theorem follows as a corollary
to the above result.
Moreover both extreme cases
in the divisibility formula can occur, in the sense that if
G w
is irreducible,
\jJG(l) == (G:H]
while in the case of the Steinberg character a Chevalley group
]E
[G:H]
of is
[1] ,
relatively prime to X(l) == 9,cm. J
X
, and
{ind x.} . ]
The theorems were motivated partly by the apparent usefulness of centralizer rings of induced representations in the problem of computing the irreducible characters of the Chevalley groups 2.
See [2],
(53.18), p. 365.
137
(see [4],
[7]).
REFERENCES 1.
Curtis, C. W., The Steinberg character of a finite group with a
(B, N)-pair, J. Algebra
4 (1966), 433-441. 2.
Curtis, C. W. and Reiner, I., Representation Theory of Finite Groups and Associative Algebras, Wiley (Interscience), New York 1962.
3.
Dade, E. C., On Brauer's second main theorem, J. Algebra 2 (1965), 299-311.
4.
Gelfand, I. M and Graev, M. I., Construction of irreducible representations of simple algebraic groups over a finite field, Doklady, 147,
(1962),529-532; Soviet Math.
3 (1962). 5.
Janusz, G., Primitive idempotents in group a1gebras, Proc. Amer. Math. Soc. 17 (1966), 520-523.
6.
Tamaschke, 0., S-rings and the irreducible representations of finite groups, J. Algebra
138
FINITE GROUPS 1 (1964),
7.
215-232.
Yokonuma, T., Sur le commutant d'une representation d'un groupe de Chevnlley fini, C. R. Acad. Sci. paris,
264 (1967), 433-436.
University of Oregon
SOME PROPERTIES OF THE GREEN CORRESPONDENCE WaIter Feit
Let
p
>
0
be a prime.
Let
R
be a
complete local domain whose maximal ideal principal such that Let
K
R/(n)
( n)
has characteristic Thus
be the quotient field of
R
either a field of characteristic
p
discrete valuation ring.
be a finite
Let
G
R
is p . is
or a complete
group. By a module we will always mean a finitely generated right module. If v
V
is an
R[G]
module and
denotes the image of
Similarly R[G]
=
-
R
= R/(n)
R[G]/(n)R[G]
v
in
V=
v
E
V
then
V/(n)V .
and ~
R[G]
We will freely use standard terminology and notation. 139
140
FINITE GROUPS Let
G.
If
S)
be
V, Ware
if there exist V
V'
~
W
z
V' m W' V
is
o
a nonempty set of subgroups of R[G]
R[G]
m W'
modules write
modules
V', W'
such that
and for every component H = H(V ) O
there exists
R[H]-projective.
S)
E
= W(S))
V
V
of
o
such that
Since the unique
decomposition property (Krull-Schmidt theorem) holds for
R[G]
modules [3,
(76.26)]
well defined equivalence relation. V
is said to be
some
x E G
If
V
I f Ax
R[~]-projective.
write
this is a
=
O(~)
E
S)
A E S) . G
Let
P
be a p-subgroup of
be a subgroup of
G
with
G
mrG(p)
C
the following sets of subgroups of
l=
l(P, H) = {AlA ~ P
x
E
~=
E
H
Define
H G.
n pX
for some
n pX
for some
G - H}
2)= 2)(P, H) = {AlA
x
and let
~
H
G - H}
~(P, H)
=
{AIA~ P
and
A
t
l}
G
Green [5] has proved the following basic resul t.
THEOREM.
There exists a function
f
from the
for
141 set of all isomorphism classes of R[G]
R[P]-projective
modules into the set of all isomorphism
classes of
R[p]-projective
R[H]
modules which
has the following properties. (i) (ii) ( iii)
f(V
(iv) (v)
Let
= f (V1)
®
V2)
-
VH (ID)
V
f(V)
2
V - o (l)
be an indecomposable ~ ;
then
a common source and vertex. R[H]
f (V )
f (V1) ® f(V 2 ) (I) f(V)G and
-
module with vertex in
posable
Gl
i f and only if
(0 )
f(V
l f (V)
V ) 2
$
l f (V)
If
V
vlw
W if and only if
In view of (v) the mapping
V (l) R[G] f (V)
have
W is an indecom-
module with vertex in G
and
-
~
then
or equivalently
f
yields a one
to one correspondence between the set of isomorphism classes of indecomposable ~
vertex in
f
with respect to If
V
moduies with
and the set of all isomorphism classes
of indecomposable We will call
R[G]
R[H]
or this the Green Correspondence (G, P, H)
is an
the submodule of
~
modules with vertex in
R[G] V
module let
InvG(V)
consisting of invariant
be
FINITE GROUPS
142
elements.
H
If
is a subgroup of
G
define
Inv (V) -+ InvG(V) by NG,H(v) I vx.l H {x. } is a (right) cross section of H
NG,H where
in
l
is the relative
G .
norm and is
(G, H)
independent of the choice of cross section. For a nonempty set and an
R[G]
Clearly
module
HO(G,~, V)
V
H
of subgroups of
G
define
is an
R
module.
In this notation D. G. Higman's criterion for relative projective modules can be formulated as follows.
THEOREM.
Let
V
be a subgroup of if and only if
be an
G
R[G]
Then
V
module and let is
H
R[H]-projective
HO(G, {H}, HomR(V, V))
=
(0)
The following result can be proved by making use of Higman's theorem. THEOREM l. let
H
Let
X, V,21
Let
P
be a p-subgroup of
be a subgroup of
G
with
be defined by (*) •
OOG(P) Let
Green correspondence with respect to
f
G C;;;;
and
H be the
(G, P, H)
143 Let
V, VI' V
R[G]
be indecomposable
2
modules.
Then
°
for any automorphism
(i)
of
R[P]-projective
R[G]
with
=
RO
R
and
GO
=
G
where
fO
is
the Green correspondence with respect to (G, pO, HO) (ii) where
If
V*
V
is R-free then
is the contragradient of
( iii)
f(Hom
(iv)
HO (G,
( v)
H (G,
R
~H
If
in a block
V
B
HO(H,
V,
Hom
(ii) that
V
HO(H,
V,
module.
~
Hom
Hom R (VI' V )) 2
H
V. (f(V ) ,f(V )) (x). l 2
V,
Hom
R
then
~
(f(V), f(V)))
BG
~
and
f(V)
is
is defined and
V
has the same meaning as in [2]).
(i) R[G] R If
K
R
f (V) *
~ HO (H, iJ), f (V))
V)
°(H,
of
COROLLARY 2. irreducible
:: Hom
has vertex in
-G B .
is in
2
x, x,
°
(vi)
(VI' V ))
=
f (V*)
If
V
module with vertex in
(f(V), f(V))) V
is an absolutely ~
=R
is an R-free
R[G]
module such
is absolutely irreducible then R
(f(V), f(V)))
then
is a cyclic
R
FINITE GROUPS
144
COROLLARY 3.
Let
p
a
Ip :AI
min
Ad that
V
is an R-free ~
and
If
dim
has vertex in irreducible.
R[G]
module such that
V 0 K K
Suppose V
is absolutely
(V 0 K)
1
of one of the following
types.
0.-.0----0
C;T)
1
O_O\'E
h
' \ Eh + l yO~o
(§)
2
I'E o_o~o
0_0
h+2t h+2t+l
Eh + t
... o_op exc Eh + t + l
155 In
the branch point
(§)
2
P
we allow
o=
Q
cannot be
Q
or
Let
D
P
but
exc
be the
set of even integers or odd integers in
M.
M (E
1
M.
Let
{O/l/ ••• /k}
1 In. )
i
E
1 lE. In. )
i
~ D
E.
'I
1
M (E.
1-
be the uniserial module
1
1-
1
1
cp.
Select homomorphisms irreducible module
1
1
FE.
E
mapping i
E
F or
D
~
i
or
D
Ei - l
~
i
F
be a monomorphism
Let
W·1
or
FE.
into
Ei - l
M.
.
In the direct sum
define the submodules
X, Y
cp.
X
1
FE.
according as
according as
1
1
D
onto the
1
or
.
D
M.
mapping
1
i
D
i
E
D
M Ell l
...
Ell M k
by (m. ) 1
and
0 < i
< k}
1
Y
1jJ. (f) 1
f
o Since all the
THEOREM.
9,(w)
if
(u r - l)a w '
(rw)
< 9, (w)
Furthermore one has
LEMMA 3.
As an
tar ; r s R}
l
2
is generated by
= ar
(for every
r s R)
u r 'a 1 + (u r - l)a r
(for every
r s R)
a a r
=
1
a a a a r s r s
-v---' m
A
together with the defining relations
alar ar
er-algebra
(for every
a a a a'" s r s r
~
in
m r,s
r,s
R
I
m
rs).
Hence one can verify easily that there exists
such that Define now
er-algebra homomorphism v(a )
for every
r
sJ sA,
d
J
s
er
r,s
being the order of
uniquely an
r,s
r
v in
as follows:
R
209
a
o
Then one can show that u d
d
OSW
.
0 J ~
Define now an {x
Then
A
{e o·
elements
Hence
J
6'-submodule
A
J
of
A
by
A
S
is a free
J
£(O)
Max
of degree equal to
c
is a polynomial in the
J
6'-module spanned by the
o S 8(J, J}}
I
a
where
T
Hence by introducing a
*
new multiplication (x
S
A
J
Y
I
S
A )
6'
~
k
we can show that
by
x
*
Y
= ~ d
(xy)
J
6'-algebra
be k-algebra homo-
defined by
respectively, where Bx
J
ep ,1/1
(r S
of the form
A
, we get an
J
Le t
morphisms
in
qr
R)
, 1/1 (u ) = 1 r
is the number of
contained in
BTI
-1
(r}B.
(r
S
R)
B-coset~
Then
210
FINITE GROUPS
Thus finally we get Theorem 1 by Lemma 1.
Let
THEOREM 2. 0
J
,
ml, ... ,m r
G, E, N, W, R, J, G , W , P , J J J
' nl, ... ,n r
be as in Theorem 1.
Suppose that every complex representation of is realized over the real number field. following conditions for a subset
Then the
of
J
W
Rare
all equivalent. is commutative,
(1 )
(2)
is commutative, (4)
n
1
=
for every
n
x
1
r
in
G, for every
( 6) (7 )
(5 )
every element of
8(J, J)
The equivalence of (1)
~
0
in
W,
is involutive. (6) is easy from
Theorem 1 and a theorem of J. S. Frame [1]. (6) is equivalent to (7) by Lemma 2. using the criterion (7) above, one can give a counter example to the following conjecture: if
G
J
is a maximal subgroup of
is commutative. (\.;,
R)
~~(G,
G,
G ) J
Namely consider the case where
is of type
(D S ) , e.g. let
G
be the
special orthogonal group on ten variables over a
711 finite field with respect to the quadratic form 10
L
X.X
i==l
1.
lO
_·
, and let
B
be the sUbgroup consisting
1.
of triangular matrices in R == {r , r , r , r 4 , r } l 3 S 2 r.r. 1.
is
3
or
2
J
We have then
G
m ..
and the order
according to
r.
and
l
of
1.J
r.
is
J
connected or not by a segment in the diagram below:
Let
== R -
J
element
{r } . 3
Then
r3r2rlr4r3rSr2r3
G(J, J)
contains the
which is not involutive.
Similarly a counter example exists for the type (F 4)
(The conjecture above is verified to be
true for types
(A ),
n
(B ),
n
(C ),
n
(G
2
)
.)
REFERENCES 1.
J. S. Frame, The double cosets of a finite group, Bull. AIDer. Math. Soc., 47, 4S8-467 (1941) .
2.
N. Iwahori, Appendix of generalized Tits system (Bruhat Decomposition) on
p -adic
212
FINITE GROUPS semi-simple groups, Proc. of Symposia in Pure Mathematics, 9 (1965).
3.
J. Tits, Theoreme de Bruhat et sous-groupes
paraboliques, C. R. Acad. Sci. 254 (1962), 3419-3422. 4.
, Algebraic and abstract simple
groups, Ann. of Math., 80 (1964), 313-329. 5.
R. Steinberg, A geometric approach to the representations of the full linear group over a Galois field, Trans. Amer. Math. Soc. 71 (1951), 274-282.
University of Tokyo
THE STEINBERG CHARACTER OF A FINITE GROUP WITH BN-PAIR Louis Solomon*
Let
1. field
F
G(q)
be a Chevalley group over the
Steinberg [3J showed that
q
G(q)
has
a remarkable irreducible character of degree equal to a power of
q
any finite group character G
~
G(q)
X
Later, Curtis [lJ showed that G
with BN-pair has an irreducible
which is equal to Steinberg's in case
, and that
X
may be written as an
alternating linear combination of characters induced from parabolic subgroups of
G.
In view
of [2, §4] one might conjecture that this formula has a homological source. is the case.
We show here that this
The requisite complex has been
defined by Tits [5, 6].
I owe a great debt to
* This work was supported in part by the National Science Foundation under grant GP-6080. 213
214
FINITE GROUPS
Tits for the lesson in "Anschauliche Geometrie" which led to the present argument.
2.
Let
rank
G
be a finite group with BN-pair [4] 1
and let
subgroups of
G , ••. , G
G
1 = VU···
U V
Q,
gG
.
i
s
V
v
is nonempty.
l G , ... ,GQ, s E 6 on
6
of dimension
6
Q,
-
g E G
G
1
A collection
if and only if
A simplex of dimension
Q,
-
1
The chamber with vertices
then
gs E 6 .
If
This action of
preserves the simplicial structure and
thus induces an action of groups of
be the
is called the fundamental chamber.
and 6
vi
g E G , and let
as its set of vertices.
is called a chamber.
G
,
of vertices is a simplex of
n VES
Let
B
The Tits complex [5, 6] of
is a simplicial complex which has
be the maximal parabolic
containing
collection of cosets V
Q,
of
G
on the homology
6
THEOREM 1.
Let
finite group
G
be the Tits complex of a
6
with BN-pair of rank
homology groups of
6
Q,
> 2
The
with integral coefficients arE
215
H.(6)=O
1
l
H (6) £-1
z e'"
~
t
Z
$
involution of the Weyl group
is the
I";
=
modulo
If
bl, ... ,b WOBW O n B
t
THEOREM 2.
Let
finite group
G
on
Q 6
then
6
a
CtL
-+
is the
£ - 1
chain
defines a
bll";, ... ,btl";
are
H£_l (6)
be the Tits complex of a
with BN-pair of rank
be the rational field.
H£_1(6) 0 Q of
W
represent the cosets
cycles which form a basis for
let
E
Wand
then the
oppositio~
LWE W E (w) wa
is a cycle. B
6
If
W
is the alternating character of fundamental chamber of
£ - 2
~
i
summands
and
where
of
2 , that i
HO (I::) " Z ,
1, .. .,£ - 1 , and
H£_l
Hi
0
w
=
p(w)
0
e
for
the representation of
(I::)
(1::)
degenerate form of Theorem 1.
e
w E W.
0
"Z
for This is a
Since
w E W , the character of W on
H£_l(l::)
is equal
to the character of the representation of
W on
and hence, by an argument given in [2, §4] is the alternating character of
W.
2]7
A glance at [3, Theorem 1] makes it apparent that this fact is a degenerate form of Theorem 2. These observations lead one to conjecture the truth of Theorems 1 and 2 on the strength of known
W.
facts about
In a conversation at Oberwolfach, Tits suggested that one could prove Theorem 1 as follows.
Follow the analogy in the preceding
paragraph.
To compute the homology of
one can choose a point
p E S£-l , let
S
9,-1
p'
antipodal point, and prove that
S£-l - {p'}
contractible by deforming it to
p
circles.
be the is
along great
Thus one should delete the chambers which are, in a precise sense,
of
opposite the fundamental chamber and prove that D - {Cl' ... ,c } is contractible. This is the t crux of the idea in the present argument although the formal proof contains less geometry and more combinatorics.
LEMMA 1.
Let
is a union
K
Suppose point,
K
be a simplicial complex which
LULl U ... U L
of subcomplexes. n has the homology of a
(i)
each
L.
(i i)
each
L n L.
l
l
has the homology of a
FINITE GROUPS
218
point, and Then
K
(iii)
and
Let
c
s
c
n c'
d(c)
let
such that
c
i
w r
LEMMA 2.
If
( i)
and
c. l
If
1
~
(ii)
If
w E W
c E r
Let
then
c ' c + i i l
cO,c1,···,c n
let
£(w)
be the
sEt!..
c E
r (s)
and
c
for some
Then
E
Co E r(s)
such that
r (s)
d(c) = d (co) + n
co,cl'···,c n = c
are adjacent and
i=O, ... , n - l .
= bwa
c
= £(w)
d(c)
for all
d (c)
there exist
=
are adjacent for
There exists a unique
d(c O)
a
, ... ,r£ .
w E W , and
LEHMA 3.
If
as a word in the distinguished
generators
b E B,
.
be the least nonnegative integer
O, ... , n - l
length of
are
£ - 2
has dimension
t!.
c E r
c, c' E r
for which there exist chambers
n
=
r
denote the set of Say that
;;;2
adjacent if E
r (s)
let
such that
c
have isomorphic homology groups.
denote the set of chambers of
r
s E t!.
If
L
E
r (s)
d(c ) i
=
then
such that d(c O) + i
for
219
Lemma 2 follows from a theorem of Tits [5, proposition 2].
Lemma 3 requires Lemma 2 and
the distinguished coset representatives [2, §3] for
W modulo the parabolic sUbgroup determined
naturally by
s.
r
ranges over set of with
Let
s E 6
If
0 < k < m
Then
=
6k- l ~ 6k
and
If
o
where
< k