Chapter 9: Counting Blocks With a Given Defect Group

Chapter 9

Counting Blocks With a Given Defect Group Our aim in this chapter is t o prove a remarkable result due t o Robinson (1983) which gives a precise formula for the number of p-blocks with a given defect group D. Although Robinson assumed D to be a normal subgroup of G, an easy modification of his proof shows that this assumption is unnecessary. Of course, Robinson was well aware of this fact. Subsequently, he provided a very elegant proof which avoided heavy computations (see Robinson (1987)). Although Robinson's theorem can be easily derived from Corollary 6.3.5, we decided t o provide a direct proof which avoids the theory of bilinear forms on G-algebras. There are a number of such proofs and we have chosen a proof due t o Kiilshammer (1984). We also provide a number of applications.

1

Preliminary results

In what follows, F denotes an arbitrary field of characteristic p > 0 and G a finite group. Given a subset X of G, we write X + E F G for the sum of a l l elements in X. If P is a p-subgroup of G, then Ip(G) denotes the F-linear span of all C + , where C is a conjugacy class of G whose defect group is G-conjugate to a subgroup of P. Recall that, by Proposition 7.1.10, Ip(G) = ( F G ) ~and so Ip(G) is an ideal of Z(FG). As before, Ib(G) denotes the F-linear span of all C + , where C is a conjugacy class of G

516

Counting Blocks With a Given Defect Group

whose defect group is G-conjugate to a proper subgroup of P. Since

we see that Ib(G) is an ideal of Z(FG). We write Gp! for the set of all p-regular elements of G and FG,, for the F-linear span of G,,. The map

designates the natural projection, i.e.

We also write c E F G for the sum of all p-elements of G (including 1). Owing to Lemma 7.7.9, there is a unique F-subalgebra A(Z(FG)) of Z(FG) such that

as F-spaces. In what follows,

denotes the natural projection. Lemma 1.1. Let D be a p-subgroup of G and let e l , . . . ,en be all block idempotents of F G with a subgroup of D as a defect group. Then (i) e l , . . . ,en are all primitive idempotents in the ideal ID(G)nA(Z(FG)) of A ( w ' G ) ) (ii) 6(ID(G)) = ID(G) n A(Z(FG)) = Cy=+(Z(FG))ei.

Proof. (i) Let e be a primitive idempotent of ID(G)rlA(Z(FG)). Then e is obviously a primitive idempotent of A(Z(FG)). Since A(Z(FG)) contains all central idempotents of F G (Theorem 7.7.10(ii)), it follows that e is a block idempotent of FG. Because e E ID(G), Proposition 7.1.6 tells us that e E {el,. . . ,en). Conversely, by assumption, each e; E ID(G) and hence

1 Preliminary results

517

by Theorem 7.7.10(ii). B;ut then each ei is a primitive idempotent of ID(G)n A(Z(FG)). (ii) Set I = I D ( G ) n A(Z(FG)). Because A(Z(FG)) is a direct sum of fields and I is an ideal of A(Z(FG)), the second equality follows by applying property (i). Taking into account that .ir(ID(G)) C ID(G), it follows from Theorem 7.7.10(i) that

On the other hand, we also have

which shows that I C ~ ( I D ( G ) )W. Let D be a p-subgroup of G. Then we denote by VD(G) the F-linear span of all C + , where C is a p-regular conjugacy class of G with D as a defect group. The next lemma illustrates how the F-space VD(G) can be brought into the argument. Lemma 1.2. Let D be a p-subgroup of G and let e be the sum of all block idempotents of F G with D as a defect group (put e = 0 if there are no such block idempotents). Then

Proof. To simplify the notation, we denote by X and Y the left and right-hand side spaces, respectively. Owing to Theorem 7.7.10(ii), we have e E A(Z(FG)). Hence

which shows that dimFX = dimFA(Z(FG))e. On the other hand, by Lemma 1.1,

Bearing in mind that

Counting Blocks With a Given Defect Group

we deduce that

Using the argument of the proof of Lemma l.l(ii), we have S(IL(G)) Ib(G). Thus the right-hand side of space (1) is the F-linear span of the elements S(C+) t Ib(G), where C is a conjugacy class of G such that D is a defect group of C. By Theorem 7.7.10(i) and Lemma l.l(ii), S(C+) E FG,, n ID(G) and therefore

for some pregular classes C; with D as a defect group and some This ensures that [S(ID(G)t Ib(G)I/Ib(G) = Y

Xi

E F.

which in turn implies that

as desired. In what follows, Sylp(G) denotes the set of all Sylow p-subgroup of G and Cl(G) denotes the set of all conjugacy classes of G. We need the following elementary group-theoretic fact.

Lemma 1.3. Let Q be a p-subgmup of G, let S be a p-regular section of G and let C be a unique p-regular class of G contained in S. Then (i) The number r of Sylow p-subgroups of G containing Q satisfies r

= 1 (mod p)

Proof. (i) The group Q acts on the set Sylp(G) by conjugation. If P E Syl,(G), then P Q if and only if {P) is a Q-orbit. Because the size of each nontrivial Q-orbit is divisible by p, we have

>

On the other hand, by Sylow's theorem ISylp(G)I l(mod p), as asserted.

2

l(modp). Thus r

=

1 Preliminary results

519

.

(ii) Let x E S and let r, be the number of Sylow p-subgroups of CG(xPl) containing < x, >. Then r, is the coefficient of x in the right-hand side of l(modp) and so the above coefficient is 1. Because (ii). Owing to (i), r, g y E S the desired assertion is for any g E C and y E P E Sylp(CG(g)), therefore established. We now apply the above lemma t o prove our final result of this section. Lemma 1.4. Let S be a p-regular section of G and let Co be a vnique p-regular class of G contained in S . Fix s E Co, P E Sylp(CG(s))and, for any given C , K E Cl(G),put

and write IPS+ =

(ass E F)

oRsC+ CcCl(G)

Then X(C, K ) is an integer such that

Proof.

Choose co E C E Cl(G). Invoking Lemma 1.3(ii), we have

On the other hand, it is easy to see that

ks'x = c0)I = I{(k,sl,Q,x)lk E K7s' E CO,XE Q E S y l P ( C ~ ( s t ) ) ,

ks'x E C)I/ICI = [ { ( c ,k,sl,Q)lc E C , k E K,st

E Co,Q E S y l p ( C ~ ( s l ) ) ,

c-'ks' E Q}I/ICI = lCol ISyl,(Cc(s))l I{(c,k) E C

x ~ l c - l k sE P)I/ICI

Counting Blocks With a Given Defect Group

520

Because the above number is an integer and

we conclude that X(C, K) is an integer such that

as desired. H

2

Robinson's theorem

In this section, we provide a proof of Robinson's theorem (1983, 1987) based on a work of Kiilshammer (1984). The corresponding result requires a fair amount of notation. For this reason, it will be convenient to assemble most of it a t the beginning. Throughout this section, F denotes an arbitrary field of characteristic p > 0, D a p-subgroup of G of order pd, P is a Sylow p-subgroup of G of order pa and C1,. . . ,Cm are all p-regular conjugacy classes of G with defect group D. We choose {g1,g2,. . . ,gr) to be a full set of (P, P)-double coset representatives in G. We also put

where i E (1,. . . ,T ) and j E { I , .. . ,m}, and introduce

N = (nij) E MTXm(E',),N~ and tij =

{I { ( x ,

the transpose of

Y) E Ci x C j 1 x - l ~E

N

~ ) l / ~ ~ -. IF ~ )

where i , j E { I , . . . ,m). By Lemma 1.4 applied t o S+ = c, where c is the sum of all pelements of G including 1, K = Cj and C = C;, the number in parenthesis is an integer. Hence we may introduce

Finally, all notation and conventions introduced in Sec.1 remain in force. We are now ready t o prove

2 Robinson's theorem

521

Theorem 2.1. Let e be the sum of all block idempotents of F G with defect group D and put e = 0 if there are no such block idempotents. Denote b y n o the number of blocks of F G with defect group D. Then

In particular, if F is a splitting field for Z ( F G ) , then

Proof. Let c be the sum of all p-elements of G including 1, fix j E be the coefficient of C,f in CTe. By Lemma l.l(ii), 6 ( C T ) E I D ( G ) and, by Theorem 7.7.10(i), 6(C:) = rr(CTc). Thus

11,. ..,m ) and let a,

m

6 ( C T )E

(mod I b ( G ) ) i=l

Applying Lemma 1.4 for S+ = c, li = Cj and C = C; we also have

We now claim that the result will be established, provided we show that

TOthis end, we set

m

M' = (a,),

uj

=

Ca,~? i=l

and

w = [ ~ ( V D ( G+)I b ( G ) ] I I b ( G ) ( I ) , W is the F-linear span of {vl + I b ( G ) ,. . . ,V m + I ) ( G ) ) .

Owing t o Denote by V the F-linear span of { q , .. . ,vm). Because V C V D ( G )and V D ( G )n I b ( G ) = 0, it follows that dimFV = dirnFW = rank(M1). By ( 2 ) and (3), we also have

Counting Blocks With a Given Defect Group

522

Applying Lemma 1.2, it follows that

a s claimed. By the foregoing, we are left t o verify (3). To this end, suppose that X P PgP so that X P = ygP for some y E P. Then

c

Applying (2) together with the fact that PgP contains exactly

left cosets, we obtain

.

Now fix k E (1,. . ,r}. Then the group ( P n S k ~ g x (iPln)g on ( g k Pn Ci) x (gkP n Cj) by the rule (a, b)("yy)= ( x - l a x , ydlby) (a E gkP

n C;, b E gkP n Cj, x, y E P n g

k ~ g p because )

and Y - ' ~ Y E gkpgkl(gkP) = gkP Note also that ( a ,b)("4)= ( a ,b) if and only if

and

Y E C~(b)n P n g k ~ g k=l C p ( b ) Consequently, the size 1 of the orbit of (a, b ) is given by

(4)

k ~ ~ kacts l )

3 The F-dimension of J ( Z ( F G ) )

But, if either Cp(a)

ft' Sylp(CG(a))or Cp(b) # P n g k ~ g , l , then

On the other hand, if Cp(a) E S y l , ( C ~ ( a ) ) and Cp(b) = P n g k P g i l , then

Consequently, we must have

and

P n gkpgkl = C P ( ~ E) S y l p ( C ~ ( b ) ) Thus in this case we have 1 = 1 and

Applying (4), we therefore deduce that

This proves (3) and the result is established.

3

The F-dimension of J(Z(FG))

In what follows, F denotes a n arbitrary field of prime characteristic p and D is a p-subgroup of G. Our aim is t o investigate the F-dimension of J ( Z ( F G ) ) by applying Theorem 2.1. Let us fix the following notation. Denote by C1,. . . ,C, all p-regular conjugacy classes of G having the same defect group D and let e o be the sum of d l block idempotents of F G with defect group D. By convention, eD = 0 if there are no such block idempotents. We fix a Sylow p-subgroup P of G and define the matrix MD = (tij) E M,(Fp) as follows :

524

Counting Blocks With a Given Defect Group

. . ( 1 Z , J 5 m), where pa and pd are the orders of P and D , respectively. As has been observed in Sec.2, the number in parenthesis is an integer and therefore MD E M , (IFP). First, we provide the following characterization of d i m F J ( Z ( F G ) ) .

<

T h e o r e m 3.1. Let r be the number of conjugacy classes of G and let X be the set of all nonisomorphic defect groups of p-regular conjugacy classes of G . Then d i m F J ( Z ( F G ) )= r rant(MD) DEX

P r o o f . By Proposition 7.1.6, our choice of X guarantees t h a t X contains all nonisomorphic defect groups of blocks of FG. Thus

and therefore

Hence, taking dimensions of both sides and applying Theorem 2.1, we have

which proves the desired assertion. 4 In order to provide further information about d i m F J ( Z ( F G ) ) ,it is desirable t o investigate the rank of the matrix MD, where MD is defined prior t o Theorem 3.1. In general, this is a difficult task and no precise formula for the rank of MD in terms of the group structure of G has been discovered so far. It is quite clear that any optimism in solving this problem must be guarded. Nevertheless it is quite possible t o provide some important information, which we present below. A conjugacy class C of G is said t o be n i l p o t e n t if C+ is a nilpotent element of Z ( F G ) . T h e o r e m 3.2. Let mD be the number of non-nilpotent p-regular conjugacy classes of G with defect group D and let lD be the number of conjugacy classes of G which are contained in O,,(G) and have defect group D . Then

3 The F-dimension o f J(Z(FG))

..

525

Proof. Let C1,. , C, be all pregular conjugacy classes of G with defect group D. Then [S(VD(G)) I b ( G ) ] / I b ( G ) is the F-linear span of

+

If C: is nilpotent, then 6(C;) is a nilpotent element of A ( Z ( F G ) ) . But A ( Z ( F G ) )is semisimple and so 6(C:) = 0. Consequently,

Invoking Lemma 1.2 and Theorem 2.1, we deduce that rank(&) 5 mD. Let C1,Cz,... 1 = ID, be all conjugacy classes of G contained in Opl(G) and having defect group D. For each i E (1,. . . ,I), fix yi E C;. Suppose that x is conjugate t o y;, y is conjugate to yj and

,el,

Because x, y E Opl(G), x-'y is p-regular. Hence x-'y = 1, so x = y. This demonstrates that

Hence tij = 0 for i

so t;;

# 0 as

# j.

Now ti;, 1

< i 5 I, is the residue (modp) of

ID1 = ICG(y;)lp. Consequently, rank(MD)

> 1, as desired.

We now apply Theorems 3.1 and 3.2 t o provide the following general information concerning d i m F J ( Z ( F G ) ) .

Theorem 3.3. Let r be the number of conjugacy classes of G, let rl be the number of non-nilpotent p-regular conjugacy classes of G and let ~2 be the number of conjugacy classes of G contained in Op,(G). Then

Proof. First we apply Theorem 3.2 to deduce that

Counting Blocks With a Given Defect Group

526

where X is the set of all nonisomorphic defect groups of pregular conjugacy classes of G and mg and ID are defined in Theorem 3.2. Invoking Theorem 3.1, we infer that

mg is the number of non-nilpotent p-regular conjugacy classes But EDEX of G, while EDEX ID is the number of conjugacy classs of G contained in Opt(G). Consequently,

as desired. H

Corollary 3.4. Let G be p-nilpotent. Then d i m F J ( Z ( F G ) ) is equal to the number of p-singular conjugacy classes of G. Proof. Because G is p-nilpotent, all p-regular conjugacy classes of G are contained in O,t(G) and therefore are non-nilpotent. Thus, in the notation of Theorem 3.3, rl = r2 and r - rl is the number of p-singular conjugacy classes of G. Hence the desired conclusion follows from Theorem 3.3. 1

4

Existence of p-blocks with a given defect group

In what follows, G denotes a finite group and F an arbitrary field of characteristic p > 0. Our principal goal is to exhibit situations where grouptheoretic information can be used to demonstrate the existence of p-blocks with a given defect group. The main results are Theorems 4.1 and 4.5 below. The case where D is a Sylow p-subgroup of G is due t o Brauer and Nesbitt (1941), while the rest generalizes theorems due t o Gow (1978), Kawada (1966) and Michler (1972a). A related result was also proved by Bovdi (1961).

Theorem 4.1. Let D be a p-subgroup of G, let n g be the number of blocks of F G with defect group D ,let m g be the number of non-nilpotent pregular conjugacy classes of G with defect group D and let ED be the number of conjugacy classes of G contained in Opt(G) and having defect group D . Then n o mD

<

4

Existence of p-blocks with a given defect group

Moreover, if F is a splitting field for Z ( F G ) , then

Proof. Let eD be the sum of all block idempotents of F G with defect group D and let MD = M in the notation of Theorem 2.1. Then no

5 d i m F Z ( F G ) e D/ J(Z(FG))eD = rank(MD)

5

mD

( b y Theorems 2.1) ( b y Theorem 3.2)

If F is a splitting field for Z ( F G ) , then r a n k ( M ~ = ) n o by Theorem 2.1. Hence the second assertion follows from Theorem 3.2. R As a preliminary to the next result, let us recall the following grouptheoretic facts.

Lemma 4.2. The group G has a normal p-complement under either of the following conditions : (i) A Sylow p-subgroup P of G is abelian and q { IAut(P)I for any prime q # p dividing IGI . (ii) G has cyclic Sylow p-subgroups and p is the smallest prime divisor of IGI.

Proof. Apply Corollaries 26.1.16 and 26.1.17 Vol.1. W Another group-theoretic information is supplied by the following lemma.

Lemma 4.3. Let D be a subgroup of G of order pd and let pa be the order of Sylow p-subgroups of G . Then N G ( D ) / D has a normal p-complement under either of the following hypotheses : (i) d = a - 1 and p is the smallest prime divisor of ]GI. (ii) d = a - 2, G is of odd order and p is the smallest prime divisor of IGI-

Proof. Assume that (i) holds. Then N G ( D ) / Dhas a Sylow p-subgroup P of order p. Because IAut(P)I = p- 1 and p is the smallest prime divisor of (GI,it follows from Lemma 4.2(i) that N G ( D ) / Dhas a normal pcomplement.

Counting Blocks With a Given Defect Group

528

Now assume that (ii) holds. Then N G ( D ) / Dhas a Sylow p-subgroup P of order p or p2. By the foregoing, we may assume that [PI = p2, in which case ( A u t ( P )= [ p(p - 1) if P 2 Z,z and

IAut(P)I = p ( p - 1)2(p+ 1) if P 2 Zp x Zp Let q be a prime divisor of [GI with q # p. Because G is of odd order, q > p 1 and hence in both cases q+ IAut(P)(. Thus, by Lemma 4.2(i), N G ( D ) / Dhas a normal pcomplement. W

+

We need one more group-theoretic fact. This is supplied by

Lemma 4.4. Let P be a p-subgroup o f G and let H = NG(P). Then the map C I+ C n C G ( P )is a bijection between the conjugacy classes of G with defect group P and the conjugacy classes of H with defect group P. Proof. Let C be a conjugacy class of G with defect group P. Then there exists g E C such that P is a Sylow p-subgroup of Cc(g). Hence g E C n C G ( P ) .If g' E C n C G ( P ) ,then there is a n x E G such that xgx-' ZE 9'. Because g' E C G ( P ) ,P is a Sylow p-subgroup of Cc(gf)= x C ~ ( g ) z - l .But zPx-' is one also and so there exists y E CG(gl)with yPy-' = X P X - l ; hence y-'x E H. Since y commutes with g', we have

so that g' is conjugate to g in H. Finally, because C G ( P )4 N G ( P ) ,any element of H which is H-conjugate t o g lies in C n CG(P).T h e conclusion is that C n C G ( P )is a conjugacy class of H with defect group P. Since C H C n C G ( P )is obviously an injection, we are left t o verify that if L is a conjugacy class of H with defect group P then L = C n C G ( P ) for some conjugacy class C of G with defect group P. To this end, choose h E L such that P is a Sylow psubgroup of C H ( h ) . Then h E C G ( P )and therefore L C C n C G ( P ) ,where C is the conjugacy class of G containing h. We next show that P is a defect group of C . It will then follow that L = C n C G ( P )and so the proof will be complete. Let Q be a Sylow p subgroup of CG(h)with P C_ Q. Then Q is a defect group of C . If Q # P, choose a subgroup PI of Q such that (PI : P ) = p. Then P C N G ( P )= H and PI C H ( h ) ,a contradiction to the fact that P is a defect group of L. Thus Q = P and the result follows.

4 Existence of p-blocks with a given defect group

529

We are now ready t o prove the following result. It is worth remarking that, by Lemma 7.1.24(ii), if F is a splitting field for G, then F is a splitting field for Z ( F G ) .

Theorem 4.5. Assume that F is a splitting field for Z(FG), let D be a p-subgroup of G and let pd and pa denote the orders of D and Sylow p-subgmups of G, respectively. Then, under either of the hypotheses below, the number of blocks of F G with defect group D is equal to the number of p-regular conjugacy classes of G with defect group D : (i) G is p-nilpotent. (ii) DCG(D)/D is p-nilpotent. (iii) NG(D)/D is p-nilpotent. (iv) D is a Sylow p-subgroup of G. (v) G has cyclic Sylow p-subgroups and p is the smallest prime divisor of the order of G. (vi) The Sylow p-subgroups of G are metacyclic and (p2 - 1,[GI) = 1. (vii) d = a - 1 and p is the smallest prime divisor of [GI. (viii) d = a - 2, G is of odd order and p is the smallest prime divisor of the order of G. Proof. Let n o be the number of blocks of F G with defect group D and let so be the number of p-regular conjugacy classes of G with defect group D. Keeping the notation of Theorem 4.1, it follows that if G is pnilpotent, = so. By Lemma 4.3, then so = lD = mD. Hence, by Theorem 4.1, each of the conditions (vii), (viii) implies (iii). Moreover, (vi) implies (i) (see Huppert (1967, p.437)). Since DCG(D) is a subgroup of NG(D)/D, (iii) implies (ii). Since (iv) obviously implies (iii), and (v) implies (i) (by Lemma 4.2(ii)), we are left to verify that n o = so if (ii) holds. Assume that D C G ( D ) / D is pnilpotent. If the result is true for NG(D) then by Lemma 4.4 and Theorem 8.1.2, the corresponding result is true for G. Hence we may harmlessly assume that D d G. Let C;, Cj be p-regular conjugacy classes of G with defect group D and let x E C;, y E Cj with x-ly E P E Sylp(G). Since D d G, we have D E Sylp(CG(x)) and D E SylP(Cc(y)) so that x,y E C c ( D ) . Then x-'D and y D are p-regular elements of the p-nilpotent group DCG(D)/D. Hence xy-'D is a pregular element. However, x-lyD is also a p-element, so x-ly E D , say x-ly = d for some d E D. It follows that y = xd with xd = dx which is only possible in the case d = 1. Thus x-'y = 1 and so x = y. By

Counting Blocks With a Given Defect Group

530

repeating the argument of Theorem 3.2, we deduce that r a n k ( M D ) But r a n k ( M D )= n o and so, by Theorem 4.1, n o = s o as desired. W

> so.

We now proceed to provide more information on existence of blocks of F G with a given defect group. Let x , y E G. We say that x inverts y if x - l y x = y-l. The next result extends that of Brauer and Fowler (1955) and is implicit in the work of Wada (1977). In what follows, yl, . . . ,y, are representatives of some conjugacy classes of H with defect group D.

Theorem 4.6. (Robinson (1983)). Let F be an algebraically closed field of characteristic p > 0 let D be a p-subgroup of G and let H = N G ( D ) . If p is odd and yl,. . . ,y, are involutions of H which invert no nontrivial p-element of C H ( D ) ,then F G has at least s blocks with defect group D . Proof. We keep the notation introduced prior to Theorem 2.1. By Theorem 8.1.2, we may assume that H = G. We may also assume that y; E Ci, 1 5 i 5 s. If i # j , and x is conjugate t o y;, y is conjugate to yj, then x-ly cannot lie in P, for if it did it would have odd order any yi would be conjugate to yj. Thus tij = 0 if i # j. If yi inverts no nontrivial p-element of C G ( D ) ,it is easy to see that ti; is the residue ( m o d p ) of ( G : C G ( y ; ) ) ( P: D ) , so ti; # 0 as D E S y l p ( C G ( y ; ) )Thus . r a n k ( M ) 2 s and the result follows by virtue of Theorem 2.1.

.

We now turn our attention to existence of blocks of defect zero. The following result is partially due to Tsushima (1971).

Theorem 4.7. Let G, be the set of all p-elements of G (including 1). Then the following conditions are equivalent : (i) F G has a block of defect zero. (ii) There exists g E G such that g = a6 for some a , b E Gp such that

Proof. Let c E F G be the sum of all elements of Gp. By Theorem 7.5.4, c2 is equal to the sum of all block idempotents of F G of defect zero. Thus (i) is equivalent to c2 # 0, i.e. to Supp c2 # 0. Assume that Suppc2 # 0, say g E S u p p c 2 . Then, by definition of c ,

4 Existence of pblocks with a given defect group

g = ab for some a , b E G p and the coefficient of g in c2 is

This shows that (i) implies (ii). Conversely, if (ii) holds then g E Suppc2 and so Suppc2 # 0. Hence (ii) implies (i) and the result follows. W By saying that an element x E G has defect zero (with respect t o p), we mean that p ICG(g)l. It should be pointed out that, since Suppc2 consists of some elements of defect zero (if c2 # O), any element g which satisfies condition (ii) of Theorem 4.7 is necessarily of defect zero (with respect t o the prime p). Our next aim is t o provide an application of Theorem 4.7. First, we need t o introduce some new terminology. Let x E G . Then the extended centralizer of x, written CE(x), is defined by

+

It is clear that CE(x) is a subgroup of G containing CG(x) as a subgroup of index a t most 2. Moreover, if x is not an involution and x is inverted by an element of G, then we have

We say t h a t a: is real provided it is inverted by an element of G. In case x is inverted by a n involution, we say that x is strongly real .

Lemma 4.8. Given x E G , let P ( z ) denote the number of ordemd pairs ( u , v ) of involutions of G such that uv = x. If x # 1 is not an involution, then P(x) is the number of involutions in C c ( x ) - CG(x). In particular, if ICz(x)l is even and CG(x) contains no involutions, then P(x) is odd. Proof. Suppose that x # 1 is not an involution. Let u be an involution of C&(x) - CG(x). Then ux is also an involution and x = u(ux). Conversely,

.

any representation of x as a product of involutions is of this form. T h e final assertion follows from the obvious fact that any group of even order contains a n odd number of involutions. We have now accumulated all the information necessary to prove the following result.

Counting Blocks With a Given Defect Group

532

Theorem 4.9. (Tsushima 1971)). Assume that char F = 2 and that a Sylow 2-subgroup of G is elementary abelian. Then F G has a block of defect zero if and only if G has a strongly real element of defect zero (with respect to 2). Proof. Suppose that F G has a block of defect zero. By Theorem 4.7, there exists an element x of defect zero such that x = ab for some 2-elements a , b E G. Since x has defect zero, it cannot be a 2-element, so both a and b are involutions. Bearing in mind that

we deduce that x is strongly r e d . Conversely, suppose that there exists a strongly real element x E G of defect zero. If u, v are 2-elements of G with x = uv, then u and v are involutions. Applying Theorem 4.7, we are left to verify that P(x) is odd where P(x) is defined in Lemma 4.8. The assumption that x is strongly real forces C c ( x ) to be of even order. Since x is of defect zero, CG(x) contains no involutions. We deduce therefore that the required assertion follows by virtue of Lemma 4.8. 4 As a preliminary t o our final result, let us recall the following piece of information. The F'rattini subgroup of G , written @(G), is defined to be the intersection of all maximal subgroups of G (by definition, @(G) = 1 if G = 1). Let d(G) denote the minimal number of generators of G (by definition, d(G) = 0 if G = 1). If G is a p-group and d = d(G), then G/@(G) is elementary abelian of order pd (see Lemma 13.1.8 in Vo1.2). Observe also that if )t is a p'-automorphism of the y-group G which induces the identity on G/@(G), then II,is the identity automorphism of G (see Gorenstein (1968, p.174)). T h e following theorem allows us t o assume that we are dealing with an elementary abelian defect group when we wish t o know how many pblocks have a given defect group. Theorem 4.10. (Robinson (1983)). Let F be an algebraically closed field of characteristic p > 0, let D be a p-subgroup of G and let H = N G ( D ) . Then there is a bijective correspondence between the blocks of F G with defect group D and the blocks of F ( H / @ ( D ) ) with defect group D / @ ( D ) . Proof.

By Theorem 8.1.2, we may harmlessly assume that H = G.

4 Existence of pblocks with

a given defect group

533

Let bars denote the images in G / @ ( D )and let y l , . . . ,y, be a full set of representatives of p-regular conjugacy classes of G with defect group D. Because any pregular element which acts trivially on D / @ ( D )must already centralize D, i t follows that 81, y2,. . . ,y, is a full set of representatives of p-regular conjugacy classes of with defect group D. Denote by P a Sylow p-subgroup of G. Given p-regular conjugacy classes C;, Cj of G with yi E C;, y j E C j , set

and

x!. '3 = { ( x ,g) E C* X CjlX-ly E P )

Because P is a Sylow p-subgroup of G, we see that P is a Sylow p-subgroup of G. One easily verifies that there is a bijective correspondence between the sets Xij and Xij, for all i, j E {I, . . . ,m). Thus the matrix M as defined for G in Sec.2 and the corresponding matrix M' for G are identical. Owing t o Theorem 2.1, the rank of M' is the number of blocks of FG with defect group D and the rank of M is the number of blocks of F G with defect group D. So the theorem is verified. 4