Pure and Appllad MathamatlcH A Series of Monographs and Textbooks edltCl,.,.. Sam .... .,_ I!U."berg end Hyman •••• Coh...
65 downloads
943 Views
8MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
Pure and Appllad MathamatlcH A Series of Monographs and Textbooks edltCl,.,.. Sam .... .,_ I!U."berg end Hyman •••• Coh.,Jmbla Univer&ilty. New YCr"k,
ReCIiNT TITLES
W. VICK. Homolo~ Theory: An Introduction to Algebraic Topology E. R. KOLCHIN. Oiffcrqntial Algebra and Algebraic Croups. Gu.Ai...D J. JANUSZ. Allt'ebr.aic N'umb(:r Fi~lds
JAMP'..5
CHARACTER THEORY OF FlNITE GROUPS
A. S. E . .HOJ.I..AND, Introduction to the Theory of Entire Fum;:tiQnll WAYNE R(lSbfS AND DALX V"'RB~a(l. ConVf:X Fun(ltions
A. M. OSTItOWSlU. Solution of Equatiotl$ in gl1~lidean and Banach Spa(lcl!I, Third Edition of Solutiol\ of Equations Olnd Sydtotems o( gquationg H. M. EbWAkD:S. Ritttlanll" Zeta Function SIoMUltl. En:£NBI!:RG. Autorrtat~. uoguag'lIIts, and Machinc$: Volume A and Voluml: a MORRIS HUl::scu: AND STEffJ'ItN S",.uP., Differential Equations. D,Ynarnil.':\\l Sy,tem!!, and Linear Als:ebl'a WTl.m~.L)! MAGN:t)S. Nonc:uGliue;\rt Tes$()iatil)n!! .and Their Groups FkANCOIS T'IEV~5. Huic Linear Partial Diffel'elltial ft;q ... ~tlQns W1LLIA'M M. '600TlfRY. An Introduction to Diffetetlti+l.ble Manifolds and Riemannian
I. MARTIN ISAACS Cepl!lrtm~r'lt of Mathematil!ls University of Wiaconslr:\ Madison. Wi:ttccmsin
Geometry DRAVtON
G1I.... v. HomotoD)' theory: An Introduction to AIQ'ebraic 'l'o!)Qlogy
ROllKR1' A, ADAMS.
J011N
J.
D. V.
WIDDI!.P..
Sobolev Spaces Spectr::!.l $ynth(:sis
B~NEDuro.
The Heat
EQ\1~tion
IRvING EZRA SItGAL, Mathematical CO$mOloR;\' and ..:xtragalactic Al!!tronomy J. DIEUOONNf::. Trel\tiMJ (m Analysis; Volume II, enlar~cd and (:orrt:cted printing; Volume IV Wl!.kN!UI. GR&UR, STK.PJ'l'Jl.N HAJ,.PltIUN, ... ND RAV VANS'i'ONI!:. Connectioll', Curvature, and Cohomology: Volume 111, Cohomology ot Principal Bunulc:s and H,)tnogen~u. 5paees 1. MARTIN IsAACS. Oial'aGtel' TheOf,y or fo'initc GI"OUP~ \
In preparation R fhtoWN. Ergodic Thl!!)l"r ar~J T!))10Iogicai Dynamics Ci.IV101([1 1\. 1'II.U~SD~LT" A Fin;t COUI"5e in Rational Continuum Mechanics: Volume l, General Coni;epts K. D. STkOYAN At"'{1 W, A. J. Luxe;MBuRr.. Tntroouctiotl to the Theory of Infiniteilmlls MELVYN n&}lG~II. Noulit'ii:!~rity a"~1 FUllctional Analr5i~: J..~turts (.on NonlinGr Problcma in Mathematical AnalYl'ii~ D. M. PUTrASWAMAIAII ANll JOHN I), DIXON. Modular Re~)r~~~nt:uiot\s or Fiiiit., JAM~S
Group!!
'.@
1878
. ACADEMIC: . PIU.!. !I'.+;
\
Contents
~
CoPYlUOHT
C
r9WBY ACADEMIC PJl~, INc.
AI,L IUOUTS IlllBnVlm. NO haT OP TIl15 ,"UBLICAIION MAY Ill! 1112PIlODUCJ\b OR nA)IlS~I't't'i!fi IN "'~v lfOJ.M 01\ ltV ANY MEANS, ELHC'TRONIC oa WBCIIAf'I'.CA..L, INCLVfilNO PHOTOCOPY I JtilCQIIJ)INO. OJ. ANV ItIIJ'OULATIO,.. SYOlUOI!. ANp ltBTJ.III.VAL AVSTBW. WrrHOUI' r2RUlSSIOt
13.
144
[irQhlems
Problem~
12.
12.
198
216 ,19 237 240 260
262 ZR5
ZR7 292 ,95
2.'
Character theory provides a powerful tool for proving theorems about finite groups. 1n fact.. there are some important results, such as Frobenius' theorern. for which no proof without characters is known. (Unlil fairly recentlYI Bumsidets pQr{ theorem was another outstanding example of this.) Although" significant part of this hook deals with techniques for applying characte.. to "pur." group theory. an even larger part is d.voted to the properties Of charact6rs themselves and how lhese propertic. reflect and are reflected in the .trudure of the group. The reader will need to know sorne basic finite group theory: the Sylow theorem. and how to use them and some elementary properties of permutation group. and solvable and nilpotent group•. A knowledge of additional topics such as transfer and the Schur-Za••enh.u. theorern would be helpful at II few point. but i. not .....nti.1. The other prerequisites are Oaloi. theory and some familiarity with rings.. In summary, the content of a first-year
graduate algebra course should provide sufficient preparation. Chapter 1 consists of ring theoretic prelirninaries. and Chapters 2-·6 and 8 contain the basic material ofch.racter theory. Chapter 7 is concerned with one of the mOre important techniques for the application of characters to group theory. The ernphasis in all chapters e~cept 1.9, 10, and 15 is on characters over the complex nurnbers rather than On module •• nd representations over other fields. In Chapter 9. irreducible representation. Over arbitrary fields arc considered; and in Chapter 10. thi' is specialized to subfield. of Ihe complex numbers. Chapter 15 is an introduction (and only that) to Brauer's vii
Notation
..
M.(F) Xv
Av A>
+.1:'
M(V) ".,(V) J(A)
CI(g)
t.
X'
0(0) Z(~)
lH det X
algebra of II x /I matrices over F linear transformation of the A-module V. induced by x € A {:>Ovlx E A}
regular A-module internal direct sum sec Definition 1.12 see Lemma 1.13 the Jacobson radical, Problem 1.4 eonjug-dey class of g symmetric group of dogree /I transpose of the matrix X the order of y see Definition 2.26 restriction of Xto If s.. Problem 2.3 the algebra homomorphism l..{C[G]) .... C induced by X X-I y.1
.,',,,,'
xy. the commutator
see Lemmo 4.4 and the discussion preceding it see Definition 4.20 see Definition S.J induced character stabilizer of ~ in permutation representation inertia group. Definition 6.10 xi
xii
Nota.tion
determinant.1 order of X. equals the order of det X in group of'linear characters O. the field Ok''''') 0'(0).0"(0) minimal normal subgroups of p·power index and index prime to p R[.SI'] ring of R·linear combinations of //' s;; Irr(O) R[.9']' (.95 R[y] 19(1) z O} P(O • .If') lWII" III e JI")] g" II,'. II •• II.' see the discussion following Lemma 8,18 . •~E see the discussion at beginning of Chapter 9 F(x) the field generated over ~. by the values of X m,(x) the Schur index. Definition 10, I Z(O. A). B(O. A). 11(0, A) see the discu"ion rreceding Theorem t 1.7 M(O} the Schur multiplier. Definition 11.12 A lhe group of linear characters of A Ch(019).lrr(OI8) see the discussion preceding Definition 11.23 cor.a(R) l1' for g E G c.d,(O) {x(I)lx e Irr(O)) V(x) vanishing·offsubgroup, Ei Olx(x) ,. 0) d,I.(O) derived length of 0 : \. ~ : F(O) Fitling subgroup of G biG) max(e,d,(G)) ,'. O,(G), 0,,(0) maximal normal subgroup of order a power of p. prime to p Trrs(O) the set of S·inv.rianl Xe Irr(G) (I~G) Frattini subgroup or 0 IBr( 0) the set or irreducible Brauer characters or 0 dx'l' decomposition numbers Definition 15.9 (1). the projective character associated with", e IBr(O) BI(G) the scI of p.blocks or 0 ,.", e., l. the idempolent and algebra homomorphism of Z(F[O]) associated with B ~ BI(O) . the set of defect groups for the cl ... :K b(:K) .;i' the sum of the elements of ,r in F[G] \" a.,(X') coefficient or.;f in '" 6(8) the set of derect groups for BE Bl(O) d(B) defect of B" ill(O) o(x)
n
<x
'j "
,;J ~.i .! • '
.,'
:,,:1
Algebras, modulEll!!; and representatioIlB
':1', i
,. "'I"
"fl':.'"
'" .L) ,I ,r;;I' .
:,; .,! i I,~,';~" !,: ! ~
:,' I
;;l!iCha~~r ,theory provid~a,means or aPl>lyiog ring theoretic techniques to the stud), of finite groupw. Although much of the theory can be developed ·iii:othi.r, ways; it',se.;n,smore·natural to approach characters via rinp: (or .accurately; algebrtis);;;lbe' pUrp a
Chaptet I
to which it is applied. Similarly, functions ate written on whichever side is con"enient, but the rule for function compO.ition i. always" Ig means do I first, then g." Because of this rule, in those situations where function composition is important [as in example (b)). it will usually be convenient to write functions on the ril!ht, We now return to another examph: of an algebra, the one of primary impOrtance for us; the group algebra. (c)
Let G be a finite group. Then F[G] i. the ..t of "formal" sums
n:... a,ula, e Fl. The ,tructure of an F.vector space i. given to F[G] in the
obvious way and the element of F[G] for which a, - I and a. ~ 0 if h." 9 i. identified with g. This identification embed. G into F[G] and in fact G is a basis for F[G]. One result of this identitication is to give a new meaning for
t
a,fI· We
may noW view this expression not only asa furmal8um, but also as.
an actual surn, a linear combination of the bas.is vectors.IFinally. hJ'"dcfine multiplication on F[G], we multiply the basis vectors according to their group multiplication and extend linearly to all of F[G). It is ,outineto check that this defines the structU,e of an F·algebra on F[G]. The construction of F[G)"suggests a general m.thod:~rcon.tructing algebras which should be mentioned. Let A be a finite dimenSional F-algebra with F-basis ViI •••• VII' We have then v,vJ l1li L'CIJ.kVAli where.cj~'e- F ate the, multiplication constants of A with respect to the basis {v,}. It is'cl.;ar that th.;'; constants determine I,he algebra. so that any n.-dimensional algebra may-bO° specified by prescribing n 3 constants c'ji. ~ F. Of coursc, only a small su.bset
of all possible sets of constants define an algebra since mos.t sets of consta.nts dl:flnc multiplications that turn out to be nonassociative. From now on, the word "algebra" in this book will mean a finite di"! mensional algebra. We make a ft:w observations and definitiOIl8 beror," going on to prove anything. Llates the maximality oW. Thus V ~ W U and V is completely reducible. Conversely,suppose V i. completely reducible and let S be the Sum of all of the irreducible oubmooules of V. If S < v. we may write V _ S 4- T with T '" O. By finite dimensionality, T contains an irreducible .ubmodule which is acontradiclion' since T n S - 0; I
+
I,
(1:1',1)" LEMMA '. LCUf, be 'a~' A;~odule and suppose V = V. where the V, are;irreducible 8ubmollules, Then V is the direet sum of SOme of the V.I 's.
L
,
"
•
, '"
i"
•
. Proof Choose w, 50' V maximal with the property that W is the direet 8um'of some' V,'8. If W '
Conversely. il X is any character of G afrorded by a represenlation X Correspondingtoamodule V. we can decompo.e V inlo a directsl,m ofi"'educiblc modules. It follows that X is the Sum of the corresponding irreducible characters. We have. in facl. X ~ L nM,( V)X,. . Corollary I. I 7(d) asserts thaI dim(C[G)) ~ D.,(dim M,l'. Since d,m C[G] 101 and dim M, ~ deg 3', ~ xli), we obtain the fundamcntal formula
particular, the functi.on 1(:; with constant value 1 on G is a linear F.charactc,r~ It is calle
It seems nalural at this point to ask how we can determine the inleger k purely group theoretically. wilhout looking at represcnt.tions. By Corollary 1.17(e). we have k ~ 1"AI(C[a])1 ~ dim Z(C[G]l.
a
(2.3) LEMMA (a) Similar F-roprcscnlaliollll of .fford equal characters. (b) Characters arc conSlanl on lh. conjugacy cl• ...,. of. group, Proo/ If Pis nonsingularlhen triP" I A. P) .. tr(P, p" J A) ~ triA). Both (a) and (b) lollow Irom this observation. To see (b), observe that :£(h-Igh)I(W I X(glI(h) if X i. a representation of a'land, he,~,c~U~I.(h-lgh»,~ tr(X(o». I ",,,,,'!,,,, ",.' ",'; ',; "'~I
.
'"' \,
Ii:
I' ': ' ., 0;
1' ,, '' '" ! "
•
We make one further general observation, If;l,and '~t'~rc F-representlff
11on50
ra
h ,t cn
'
:.,1"".(l1""1",,,'"1':'1:""!
, ,,' , I. show that,:xo/I ¢ lrr{G). (a)
No.. (2.7)
i
Note
Show that ktr{det X) contains all elementoof ofder4.'U.c. I:emma
2.15.
(2.6)
31
•
:,' .
, ,,', !.
I;
,..',. '..
for every nonlincar X" Irr( G).
'
! r~
,
(a) Show that C i. an abeli..ngroup WIder, the' multiplication of , '" . ' , ' . ',I .' \. : , . ".' Problem 2.6. '(b) If H I. then X(I) (K,l ~ Z:a", {.'I',}. Since for each i there exists j with cjj #0 O~' this map is onto and hence is one-to-one. Because all non~ero tlJ!;:c ± I, we have L/ = ±K] when:f./ and .:;t-] coreespond.Finally.Equ.tion(l)yicldsIHII.'I'd = IGIIX"Jlandthusl2',I-IX"jl and the proof is complete. I
We need a lemma. (3.t8) Lf.ldMA Let «. Ii ~ Z(C[G]) and view the characters of G as being defined on all of C[ G]. Define
( I, we have Z ~ C1, Also G/Z is not eyclic 8it'lce G i$ not abelian. If G/2 ,.lbJi., Ih. hypotheses of the theorem, then IG:G'I = I(G/2):(G/2)'1 ~ 4 by induction and we are done. We may therefore assume that the number of involutions in O/Z is not .. I mod 4, We have :i: v,(x)x(1) t + 1 '" 2 mod 4 XoIIlu(G)
I
",(x)x( I) ;0. mod 4,
x~lrr(G);1.I:t."l
where the second (. )
'
4 ... x(1)' ;., [X" X,] - 10: Fl. ....,.,
(Alperin"Pelr- Thomp,"n) Let 0 be a 2-group containing exactly t involutions. If t ;- 1 mod 4, then either G is cyclic or IG: I ~ 4. THF,ORO..
and
..
"
I
following. (4.9)
,
where t!I is the orbit containing X. . Since Z " ker X, we have ker X ~ 1. Since G is not abelian. x(t) :> I and hence x(1) m 2 and 1t!l1-1. Therefore . .lx - X for all ,leG. Now let F = 2, Theil G contains a proper subgroup of order> (g)"'. Proof We use induction on IGI, First aSSume Z(G) - Z > l, [fIG:ZI 'i. even, then there e.iSls H/Z < G/Z with
X(I).
where
result follows.
I.}.
Now assume 7..(G)
[n particular, s _ 1.'1'1 .. O. Now I' ,.
S8
Products of charactor::!
(~X(!)), ,. s ~ ~G~'~s(g ~;/'
= I and let, be as in Theorem 4.11. Then I
< IG:C(x)1 ",'
sOme x e G·and we are done if III < , • be an involution in G, Then
by Lemma 4,10, It follows that
g"l/J.
Assume. then. that
IX ;;: () l/~
and
/
I < IG: c(.)1 ,. (g - 1)/, < yja ,. g'" the resuh follows.
; .'1
and thus
sa' ;;,;
I
,.'.
L1(1)', " go
Therefore. x(l) s • for .omc X e .'1', proving (b), " " Since k - I, where k ~ I[rr(G) I i. the total number of conjugacy c:lasscs of G~ we have .f
s"
,.1/
~"
".
and hence some nonidentity class of G has size ,:;;a~. This pr?ve.(a}.
I>';'
Recall that an element x ~ G is said to be real if x i.,conjugatein 0 to x-,\;;, It is a fact (which will be proved later) that the number of cla.se. of toal elements of G is equal to the number of reaLirreducible character•. Assuming this, it i. immediate from, the ,inequalitY' ••':;;'; g ~ ,1 inth. above proof that statement (a) can be strengthened to guarant.. that a real XEG exists with x '" landIG:C(x)l ,..'. .. I:,'" n,:, I,,: (4_ i2) COROLLARY Let n be a positive integer. There exisi at most finitely many simple groups containing an involution with centrali~r of order n. Proof Lei G be such a g"oup with IG I ~ g, Then G contains at involu~ions a.nd hence a. -:;: n in the notation of Thcorem 4.11.
least gin
HOi
:respectivcly. An clement w IS U V is uniquely of the form w=
"
L 1.1
.-
Q
,JII 1
0";I
and this defincnhc" x m matrix (a I)' We write M(w)
1'". ""mpIJle M(wg).
= (a I)' for y E G, wc
" Let l: and 'D be representations of G corresponding to U and V respectiv