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On TI and TI Defect Blocks
Journal of Algebra 243, 123᎐130 Ž2001. doi:10.1006rjabr.2000.8624, available online at http:rrwww.idealibrary.com on
On TI and TI Defect Blocks Jianbei An Department of Mathematics, Uni¨ ersity of Auckland, Auckland, New Zealand E-mail: [email protected]
and Charles W. Eaton School of Mathematics and Statistics, Uni¨ ersity of Birmingham, Birmingham, England E-mail: [email protected] Communicated by Michel Broue´ Received April 8, 2000
We show that every trivial intersection block of a finite group Žas introduced by J. L. Alperin and M. Broue ´ Ž1979, Ann. of Math. 110, 143᎐157.. has trivial intersection ŽTI. defect groups but that the converse is not true in general. We then present some conditions equivalent to B being a TI block, generalizing the idea of a k-generated p-core to B-subgroups. In particular we give further weight to Olsson’s assertion that TI blocks are a better generalization of groups with TI Sylow p-subgroups than are TI defect blocks. Finally we describe the role ˆ of the 䊚 2001 Acageneralized k-generated p-core in the control of fusion of subpairs. demic Press
1. INTRODUCTION AND DEFINITIONS Let G be a finite group, let p be a prime, and let BlkŽ G . be the set of all p-blocks of G. A subgroup H of G is called a tri¨ ial intersection Žfor short TI. set if H g l H s 1 for every g g G _ NG Ž H .. Let B be a p-block with a defect group D s DŽ B .. Then B is called a TI defect block if D is a TI set. A pair Ž R, bR . of a p-subgroup R F G and a block bR g BlkŽ RCG Ž R .. is called a subpair of G. Given B g BlkŽ G ., a subpair Ž R, bR . is called a B-subgroup Žsee w1x. if Ž1, B . F Ž R, bR . and is non-tri¨ ial if R / 1. If, moreover, R is a defect group DŽ B . of B, then a B-subpair Ž R, bR . is 123 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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called a Sylow B-subgroup. Following w2x, B is called a trivial intersection Žfor short TI. block if each non-trivial B-subgroup is contained in a unique Sylow B-subgroup. Some properties of TI blocks can be found in the papers of Alperin and Broue ´ w2x and Olsson w8x. Given a B-subgroup Ž Q, bQ ., we define ⌫ŽQ , b Q ., k Ž G . s² NG Ž R, bR . : Ž R, bR . F Ž Q, bG . , m p Ž R . G k: , where m p Ž R . is the p-rank of R. If Ž D, bD . is a Sylow B-subgroup, then we call ⌫Ž D, b D ., k Ž G . the k-generated B-core, which is completely determined up to conjugacy in G. We say that a proper subgroup H of G is a strongly B-embedded subgroup of G if H G ⌫Ž D, b D ., 1Ž G ., where Ž D, bD . is a Sylow B-subgroup. If B0 is the principal p-block of G, then Brauer’s third main theorem tells us that ⌫Ž D, b D ., k Ž G . is the k-generated p-core ⌫D, k Ž G ., and H is a strongly B0-embedded subgroup if and only if it is a strongly p-embedded subgroup. The structure of this paper is as follows: in Section 2 we strengthen our characterization of TI defect blocks in terms of the p-local rank of a block introduced in w3x. In Section 3 we show that if B is a TI block, then it must also be a TI defect block. We give an example Žand means of generating more. of a TI defect block which is not TI. In Section 4 we give some equivalent definitions of a TI block, and finally in Section 5 we consider the k-generated B-core and describe the part it plays in the fusion of subpairs.
2. TI DEFECT BLOCKS Before we state our result, we remind the reader of the concepts involved: Given a chain of p-subgroups : Q 0 - Q1 - ⭈⭈⭈ - Q n of G, define the length < < s n, the final subgroup V s Q n , the initial subgroup V s Q 0 , the kth initial subchain k : Q 0 - Q1 - ⭈⭈⭈ - Q k , and the normalizer NG Ž . s NG Ž Q 0 . l NG Ž Q1 . l ⭈⭈⭈ l NG Ž Q n .. A chain is said to be radical if Q i s Op Ž NG Ž i .. for each i. Denote by R s RŽ G . the set of all radical p-chains of G and write RŽ G, B . s g RŽ G . : BlkŽ NG Ž . : B . / ⭋4 , where for a subgroup H, BlkŽ H : B . s b g BlkŽ H . : b G s B4 Žin the sense of Brauer.. Following w3x, the p-local rank plrŽ B . of B is defined to be the number plrŽ B . s max< < : g RŽ G, B .4 . We improve a result of w3x in order to consider the p-local rank of a TI block later. PROPOSITION 2.1. A block B g BlkŽ G . is a TI defect block if and only if either plrŽ B . s 0 or plrŽ B . s 1 and Op Ž G . s 1.
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Proof. Suppose B is a TI defect block with a defect group D s DŽ B .. Then D is a radical subgroup of G and so Op Ž G . F D, where a subgroup R of G is radical if R s Op Ž NG Ž R ... Let G s GrOp Ž G . and denote by x the image of an element x g G in G. Then NG Ž D . s NG Ž D . and x s xOp Ž G . g G _ NG Ž D . for some x g G if and only if x g G _ NG Ž D .. It follows that if D / Op Ž G ., then DrOp Ž G . is a TI set in GrOp Ž G . and by w3, Proposition 5.1x, plrŽ B . s 1. Thus either plrŽ B . s 0 or 1. In the latter case, Op Ž G . s 1 since Op Ž G . F D l D x for each x g G. Suppose plrŽ B . s 0. Since D is radical in G, it follows that D s Op Ž G ., and since D is normal in G, it is a TI set and B is a TI defect block. Suppose plrŽ B . s 1 and Op Ž G . s 1. Then D / 1 and again by w3, Proposition 5.1x, D s DrOp Ž G . is a TI set of G s GrOp Ž G ., so that B is a TI defect block.
3. TI AND TI DEFECT BLOCKS LEMMA 3.1. Let B be a TI block Ž D, bD . a Sylow B-subgroup and Ž Q, bQ . F Ž D, bD .. Then there is a B-subgroup Ž P, bP . with Ž Q, bQ .eŽ P, bP . F Ž D, bD . such that NG Ž Q, bQ . F NG Ž P, bP .. Proof. Let P s ND Ž Q . ) Q. Then there is an unique Ž P, bP . such that
Ž Q, bQ . e Ž P , bP . F Ž D, bD . . Since B is TI, we have NG Ž Q, bQ . F NG Ž D, bD . F NG Ž D .. So NG Ž Q, bQ . F NG Ž P .. Let x g NG Ž Q, bQ .. Then Ž P, bP . x s Ž P, bPx .. But x g NG Ž D, bD ., so both Ž P, bP . and Ž P, bPx . are contained in Ž D, bP .. Hence x g NG Ž P, bP . and we are done. PROPOSITION 3.2. Let B be a TI block, and let Ž Q, bQ . be a non-tri¨ ial B-subgroup. Suppose Q is a subgroup of a defect group D s DŽ B .. Then Ž Q, bQ . is contained in a unique Sylow B-subgroup Ž D, bD .. Proof. It suffices to show the existence of the Sylow B-subgroup Ž D, bD .. Let Q be maximal in D such that there is a B-subgroup Ž Q, bQ . with no Sylow B-subgroup of the form Ž D, bD . containing it. Clearly Q - D. Let Ž D, bXD . be a fixed Sylow B-subgroup. This defines a B-subgroup Ž Q, b 0 . contained in it. Then by Lemma 3.1 there is a B-subgroup Ž P, b1 . with Ž Q, b 0 .eŽ P, b1 . F Ž D, bXD . and NG Ž Q, b 0 . F NG Ž P, b1 .. Now CG Ž Q . Q F NG Ž Q, b 0 . and NG Ž P, b1 . F NG Ž P ., so CG Ž Q . Q F NG Ž P . and C G Ž Q . P F NG Ž P . . Since C G Ž P . F C G Ž Q . and Ž . Ž . CG Ž P . P e ᎏ NG P , it follows that both CG P P and P are normal subgroups of CG Ž Q . P. If H s CG Ž Q . P, then CH Ž Q . s CG Ž Q ., Ž Q, bQ . is a
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subpair of H and bQH is a well-defined block of H. Set BH s bQH. Then BHG s B as bQG s B, and hence BH g BlkŽ H N B .. Let bP be a block of CG Ž P . P covered by BH . By w1, Theorem 15.1x, bPh s BH s bQH and so Ž Q, bQ .eŽ P, bP .. By the maximality of Q, since Q - P there is a Sylow B-subgroup Ž D, bD . containing Ž P, bP ., and so also Ž Q, bQ ., a contradiction. PROPOSITION 3.3. A TI block B is a TI defect block. In particular, plrŽ B . s 0 or 1, and in the latter case Op Ž G . s 1. Proof. Let B be a TI block and suppose D s DŽ B . is not a TI set. Then 1 / D l D x for some x g G _ NG Ž D .. Let Ž D, bD . be a fixed Sylow B-subgroup and let Q s D l D x. Then Ž Q, bQ . F Ž D, bD . for a unique block bQ g BlkŽ CG Ž Q . Q ., so that Ž Q, bQ . is a non-trivial B-subgroup. Since D x is also a defect group of B and Q F D x , it follows by Proposition 3.2 that Ž Q, bQ . - Ž D x , bx . for some Sylow B-subgroup Ž D x , bx ., which is impossible since Ž D, bD . / Ž D x , bx .. Thus B is a TI defect group, and the last assertion follows by Proposition 2.1. Suppose B s B0 Ž G . is the principal block of G. If B is a TI defect block and if a non-trivial B-subgroup Ž R, bR . is contained in Sylow B-subgroups Ž D, bD . and Ž D⬘, bD ⬘ ., then D⬘ s D x for some x g G and R F D l D⬘ s D l D x. Thus x g NG Ž D . and D s D⬘ as D is a TI set and R / 1. By Brauer’s third main theorem, bD s bD ⬘ is the principal block of CG Ž D . D, so that B is a TI block. It follows that if a principal block is a TI defect block, then it is a TI block. But this is not true in general. We describe where examples of TI defect blocks which are not TI may be found and then give a specific example. Suppose that G has TI Sylow p-subgroups and that these Sylow p-subgroups are abelian and have p-rank greater than one. Then G s Op⬘Ž G ..Ž N. A., where N is a nonabelian simple group and A is a finite group such that N. A acts faithfully on N Žsee, for example, w5, 1.2x.. Let P g Syl p Ž G . Žand note that PCG Ž P . s CG Ž P ... Then Op⬘Ž G . F NG Ž P ., and so Op⬘Ž G . F CG Ž P .. Hence N F X s ² P : P g Syl p Ž G .: e yG, and Op⬘Ž G . centralizes N. So G has the form Ž Op⬘Ž G .夹M .. A for some quasisimple group M with TI Sylow p-subgroups, where Ž Op⬘Ž G .夹M . is a central product of Op⬘Ž G . and M over Op⬘Ž G . l M. To produce examples of TI defect blocks which are not TI, it suffices to find non-abelian simple groups M with TI Sylow p-subgroups and outer automorphism groups A of M, such that for a non-trivial subgroup Q - P we have CAŽ Q . / CAŽ P .. Then we may choose Op⬘Ž G . such that CAŽ Q . and CAŽ P . act non-trivially and trivially on Op⬘Ž G ., respectively. Choose 1 , 2 g IrrŽ Op⬘Ž G .. fixed by
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CAŽ P . but conjugate in CAŽ Q ., and consider bP g BlkŽ CG Ž P .. covering 1 Žafter identifying i with the block of defect zero containing it. and its Brauer correspondent bQ in CG Ž Q .. But bQ must also cover 2 , so there must be a block bXP / bP of CG Ž P . covering 2 which is also a Brauer correspondent of bQ Žconsider the conjugate of bP by the element of x g CG Ž Q . taking 1 to 2 .. This would give the counterexample we require. In particular, consider the following subgroup of S10 . Take M s A 6 F S10 permuting 1, . . . , 6 4 and O p ⬘ Ž G . s ²Ž7, 8, 9, 10 .: . Take P s ²Ž1, 2, 3., Ž4, 5, 6.: and Q s ²Ž4, 5, 6.:. Let A s ²Ž1, 2. Ž7, 10. Ž8, 9.:, and define G s Ž M = Op⬘Ž G ... A. Note that A 6 has TI Sylow 3-subgroups, hence so has G, and every 3-block of G is TI defect. Then CG Ž P . s P = Op⬘Ž G . and so has four 3-blocks. CG Ž Q . s Ž P = Op⬘Ž G ... A and has three 3-blocks. Now every 3-block of CG Ž P . has a Brauer correspondent in CG Ž Q ., and so there must be two distinct 3-blocks bP , bXP of CG Ž P . sharing a 3-block bQ of CG Ž Q .. Hence Ž Q, bQ . is contained in distinct subpairs Ž P, bP . and Ž P, bXP ., so bQG has TI defect groups but is not TI.
4. SOME ALTERNATIVE CHARACTERIZATIONS OF TI BLOCKS A B-subgroup Ž Q, bQ . of G is special if it is maximal amongst those NG Ž Q, bQ .-stable subpairs for NG Ž Q, bQ .. ŽRecall that if Ž R, bR . is a subpair for NG Ž Q, bQ . containing Ž Q, bQ ., then it is necessarily a subpair for G and a B-subgroup.. Note that Ž1, B . and the Sylow B-subgroups are special. Note also that if B0 is the principal p-block, then the special B-subgroups correspond to the radical p-subgroups of G. Before proving our main result we give a well-known but useful lemma: LEMMA 4.1. Let Ž Q, bQ . be a B-subpair, where B is a p-block of a finite group G, and let Ž P, bP . be a B-subpair properly containing Ž Q, bQ .. Then Q - NP Ž Q, bQ .. Proof. Write R s NP Ž Q . ) Q. Then there is a B-subpair Ž R, bR . with Ž Q, bQ . F Ž R, bR . F Ž P, bP .. Since Qe R, it follows that Ž Q, bQ .eŽ R, bR ., where by definition Q - R F NG Ž Q, bQ ., and the result follows. PROPOSITION 4.2. Let B be a p-block of a finite group G with Op Ž G . s 1. Then the following are equi¨ alent: Ži. B is a TI block. Žii. NG Ž D, bD . is a strongly B-embedded subgroup for each Sylow B-subgroup Ž D, bD ..
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Žiii. The only non-tri¨ ial special B-subgroups are the Sylow B-subgroups. Proof. Ži. « Žii. is immediate. Žii. « Žiii. Suppose that Ž Q, bQ . is a non-trivial special B-subgroup. Then Ž Q, bQ . F Ž D, bD . for some Sylow B-subgroup Ž D, bD .. Suppose that Q / D. Then R s ND Ž Q, bQ . ) Q by Lemma 4.1. Let bR be the unique p-block of RCG Ž R . such that Ž Q, bQ . - Ž R, bR . F Ž D, bD .. Then CG Ž R . F CG Ž Q . F NG Ž Q, bQ . and R F NG Ž Q, bQ ., so Ž R, bR . is a subpair for NG Ž Q, bQ .. But NG Ž R, bR . F NG Ž D, bD . by our assumption, so Ž R, bR . is NG Ž Q, bQ .-stable, since NG Ž Q, bQ . F NG Ž D, bD .. Hence Ž R, bR . is a NG Ž Q, bQ .-stable subpair for NG Ž Q, bQ . which strictly contains Ž Q, bQ ., contradicting our assumption that Ž Q, bQ . is special. So Q s D and we are done. Žiii. « Ži. Suppose that B is not a TI, but that condition Žiii. holds. Choose Q maximal for which there is some bQ g BlkŽ QCG Ž Q . ¬ B . such that Ž Q, bQ . is contained in two distinct Sylow B-subgroups Ž D, bD . and Ž D 1 , bD .. Then Ž Q, bQ . is nontrivial, a B-subgroup, and not special. 1 Consider a chain
Ž Q, bQ . s Ž Q0 , b0 . F Ž Q1 , b1 . F Ž Q2 , b2 . F
⭈⭈⭈
of B-subgroups, where Ž Q iq1 , biq1 . is a maximal NG Ž Q i , bi .-stable subpair for NG Ž Q i , bi . containing Ž Q i , bi .. Since G is finite eventually there must be an n such that
Ž Q nq j , Q nqj . s Ž Q n , bn . for every j G 0. But then Ž Q n , bn . is special, and so it is a Sylow B-subgroup, say Ž D⬘, bD ⬘ .. Note that by definition NG Ž Q 0 , b 0 . F NG Ž Q1 , b1 . F ⭈⭈⭈ F NG Ž Q n , bn ., so NG Ž Q, bQ . F NG Ž D⬘, bD ⬘ .. Without loss of generality we may take Ž D⬘, bD ⬘ . s Ž D, bD .. Let R s ND 1Ž Q, bQ ., so Q - R F D 1 by Lemma 4.1. Let bR be the unique block of BlkŽ RCG Ž R .. such that Ž R, bR . F Ž D 1 , bD 1 .. We have CG Ž R . F CG Ž Q . F NG Ž Q, bQ . F NG Ž D, bD . , and also R F NG Ž Q, bQ . F NG Ž D, bD ., so Ž R, bR . is a subpair for NG Ž D, bD . and bRNG Ž D, b D . is defined. But Ž bRNG Ž D, b D . . G s bRG s B, so Ž D, bD . is the unique Sylow bRNG Ž D, b D .-subpair and so contains Ž R, bR .. Hence Ž R, bR . is a B-subgroup contained in distinct Sylow B-subgroups Ž D, bD ., Ž D 1 , bD 1 ., contradicting our choice Q. Remarks. Ži. It is easy to see that if B is a TI defect block, then G possesses a strongly B-embedded subgroup Žfor if Ž D, bD . is a Sylow B-subpair for a TI defect block B, and Ž Q, bQ . F Ž D, bD ., then NG Ž Q, bQ .
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F NG Ž Q . F NG Ž D . - G .. However, this does not lead to a strategy for proving that every TI block is TI defect by way of Proposition 4.2, for it is not necessarily the case that if G possesses a strongly B-embedded subgroup then B is a TI. For example, consider the principal 5-block of M22 Žsee w7, 24.2x.. Žii. Special subpairs may be seen as a generalization of radical p-subgroups. Define a special chain of subpairs and the special local rank analogously to the p-local rank, but use special subpairs instead of radical p-subgroups Žso a chain : Ž Q 0 , b 0 . - ⭈⭈⭈ - Ž Q n , bn . of B-subgroups is special if Ž Q 0 , b 0 . is special and Ž Q iq1 , biq1 . is a special subpair of NG Ž i . for each i .. It follows from Propositions 2.1 and 4.2 that if B is a p-block of special local rank one of a finite group with Op Ž G . s 1, then plrŽ B . s 1. 5. THE STRONGLY k-EMBEDDED B-CORE Just as ⌫P, k Ž G . plays a role ˆ in the fusion of p-subgroups Žsee w4x., we demonstrate that ⌫Ž D, b D ., k Ž G . plays a similar part in the fusion of certain B-subgroups: LEMMA 5.1. Suppose that H F G, b g BlkŽ H ¬ B ., and Ž D, bD . is a Sylow b-subgroup of H which is also a B-subgroup of G. If ⌫Ž D, b D ., k Ž G . F H, then Ž D, bD . is a Sylow B-subgroup of G. Proof. Suppose that Ž D, bD . is not a Sylow B-subgroup. Since Ž D, bD . is a B-subgroup, it follows from Lemma 4.1 that there is a B-subgroup Ž P, bP . with Ž D, bD .eŽ P, bP . and D / P, where P is contained in a defect group for B. But PCG Ž P . F PCG Ž D . F NG Ž D, bD . F H, and so Ž P, bP . is a subpair for H and necessarily a b-subgroup, contradicting our choice of Ž D, bD . as a Sylow b-subgroup. PROPOSITION 5.2. Suppose that H F G, b g BlkŽ H ¬ B ., and Ž D, bD . is a B-subgroup which is also a Sylow b-subgroup. Write SB for the poset of B-subgroups, and SB Ž b, k . s Ž Q, bQ . g SB : QCG Ž Q . F H , bQH s b, m p Ž Q . G k 4 . Then ⌫Ž D, b D ., k Ž G . F H if and only if H controls fusion in SB Ž b, k . and NG Ž Q, bQ . F H for e¨ ery Ž Q, bQ . g SB Ž b, k . and e¨ ery B-subgroup Ž Q, bQ . F Ž D, bD .. Proof. Suppose that ⌫Ž D, b D ., k Ž G . F H. Then by Lemma 5.1 Ž D, bD . is a Sylow B-subgroup. Let Ž Q, bQ . g SB Ž b, k ., and suppose that g g G satisfies Ž Q, bQ . g g SB Ž b, k .. We show that g g H, from which it follows immediately that H controls fusion in SB Ž b, k . and NG Ž Q, bQ . F H for
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every Ž Q, bQ . g SB Ž b, k .. Clearly also NG Ž Q, bQ . F H for every Ž Q, bQ . F Ž D, bD .. Since all Sylow b-subgroups are conjugate in H, there are h1 , h 2 g H such that Ž Q, bQ . h1 , Ž Q, bQ . g h 2 F Ž D, bD .. Let x s hy1 1 gh 2 g G. We may now apply w2, 4.10x Žnote that we may use the results of w2x here despite the differing definitions of a subpair, since there is a 1᎐1 correspondence between p-blocks of QCG Ž Q . and p-blocks of CG Ž Q . which respects the Brauer correspondence and conjugacy in G .: there are B-subgroups
Ž R1 , b1 . , . . . , Ž R n , bn . , all contained in Ž D, bD ., and x i g NG Ž R i , bi ., 1 F i F n, such that Ž Q, bQ . h1 F Ž R1 , b1 ., Ž Q, bQ . h1 x 1 ⭈⭈⭈ x i F Ž R iq1 , biq1 ., for i - n, and x s x 1 ⭈⭈⭈ x n . Manifestly m p Ž R i . G m p Ž Q . G k for each i, so each x i g NG Ž R i , bi . F ⌫Ž D, b D ., k Ž G . F H, and so x g H. Hence g g H as required. Now suppose that H controls fusion in SB Ž b, k . and NG Ž Q, bQ . F H for every B-subgroup Ž Q, bQ . F Ž D, bD .. Then Ž Q, bQ . g SB Ž b, k . for all such Ž Q, bQ ., and H controls fusion of B-subgroups contained in Ž D, bD .. Then it follows immediately from w2, 4.24x that NG Ž Q, bQ . s H l NG Ž Q, bQ . for each such Ž Q, bQ ., and so by definition ⌫Ž D, b D ., k Ž G . F H. REFERENCES 1. J. L. Alperin, ‘‘Local Representation Theory,’’ Cambridge Univ. Press, Cambridge, UK, 1986. 2. J. L. Alperin and M. Broue, ´ Local method in block theory, Ann. of Math. 110 Ž1979., 143᎐157. 3. J. An and C. W. Eaton, The p-local rank of a block, J. Group Theory, 3 Ž2000., 369᎐380. 4. M. Aschbacher, ‘‘Finite Group Theory,’’ Cambridge Univ. Press, Cambridge, UK, 1986. 5. H. I. Blau and G. O. Michler, Modular representation theory of finite groups with T.I. Sylow p-subgroups, Trans. Amer. Math. Soc. 319 Ž1990., 417᎐468. 6. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ‘‘An ATLAS of Finite Groups,’’ Clarendon, Oxford, 1985. 7. D. Gorenstein and R. Lyons, On finite groups of characteristic 2-type Mem. Amer. Math. Soc. 276 Ž1982.. 8. J. B. Olsson, On subpairs and modular representation theory, J. Algebra 76 Ž1982., 261᎐279.
On TI and TI Defect Blocks Jianbei An Department of Mathematics, Uni¨ ersity of Auckland, Auckland, New Zealand E-mail: [email protected]
and Charles W. Eaton School of Mathematics and Statistics, Uni¨ ersity of Birmingham, Birmingham, England E-mail: [email protected] Communicated by Michel Broue´ Received April 8, 2000
We show that every trivial intersection block of a finite group Žas introduced by J. L. Alperin and M. Broue ´ Ž1979, Ann. of Math. 110, 143᎐157.. has trivial intersection ŽTI. defect groups but that the converse is not true in general. We then present some conditions equivalent to B being a TI block, generalizing the idea of a k-generated p-core to B-subgroups. In particular we give further weight to Olsson’s assertion that TI blocks are a better generalization of groups with TI Sylow p-subgroups than are TI defect blocks. Finally we describe the role ˆ of the 䊚 2001 Acageneralized k-generated p-core in the control of fusion of subpairs. demic Press
1. INTRODUCTION AND DEFINITIONS Let G be a finite group, let p be a prime, and let BlkŽ G . be the set of all p-blocks of G. A subgroup H of G is called a tri¨ ial intersection Žfor short TI. set if H g l H s 1 for every g g G _ NG Ž H .. Let B be a p-block with a defect group D s DŽ B .. Then B is called a TI defect block if D is a TI set. A pair Ž R, bR . of a p-subgroup R F G and a block bR g BlkŽ RCG Ž R .. is called a subpair of G. Given B g BlkŽ G ., a subpair Ž R, bR . is called a B-subgroup Žsee w1x. if Ž1, B . F Ž R, bR . and is non-tri¨ ial if R / 1. If, moreover, R is a defect group DŽ B . of B, then a B-subpair Ž R, bR . is 123 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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called a Sylow B-subgroup. Following w2x, B is called a trivial intersection Žfor short TI. block if each non-trivial B-subgroup is contained in a unique Sylow B-subgroup. Some properties of TI blocks can be found in the papers of Alperin and Broue ´ w2x and Olsson w8x. Given a B-subgroup Ž Q, bQ ., we define ⌫ŽQ , b Q ., k Ž G . s² NG Ž R, bR . : Ž R, bR . F Ž Q, bG . , m p Ž R . G k: , where m p Ž R . is the p-rank of R. If Ž D, bD . is a Sylow B-subgroup, then we call ⌫Ž D, b D ., k Ž G . the k-generated B-core, which is completely determined up to conjugacy in G. We say that a proper subgroup H of G is a strongly B-embedded subgroup of G if H G ⌫Ž D, b D ., 1Ž G ., where Ž D, bD . is a Sylow B-subgroup. If B0 is the principal p-block of G, then Brauer’s third main theorem tells us that ⌫Ž D, b D ., k Ž G . is the k-generated p-core ⌫D, k Ž G ., and H is a strongly B0-embedded subgroup if and only if it is a strongly p-embedded subgroup. The structure of this paper is as follows: in Section 2 we strengthen our characterization of TI defect blocks in terms of the p-local rank of a block introduced in w3x. In Section 3 we show that if B is a TI block, then it must also be a TI defect block. We give an example Žand means of generating more. of a TI defect block which is not TI. In Section 4 we give some equivalent definitions of a TI block, and finally in Section 5 we consider the k-generated B-core and describe the part it plays in the fusion of subpairs.
2. TI DEFECT BLOCKS Before we state our result, we remind the reader of the concepts involved: Given a chain of p-subgroups : Q 0 - Q1 - ⭈⭈⭈ - Q n of G, define the length < < s n, the final subgroup V s Q n , the initial subgroup V s Q 0 , the kth initial subchain k : Q 0 - Q1 - ⭈⭈⭈ - Q k , and the normalizer NG Ž . s NG Ž Q 0 . l NG Ž Q1 . l ⭈⭈⭈ l NG Ž Q n .. A chain is said to be radical if Q i s Op Ž NG Ž i .. for each i. Denote by R s RŽ G . the set of all radical p-chains of G and write RŽ G, B . s g RŽ G . : BlkŽ NG Ž . : B . / ⭋4 , where for a subgroup H, BlkŽ H : B . s b g BlkŽ H . : b G s B4 Žin the sense of Brauer.. Following w3x, the p-local rank plrŽ B . of B is defined to be the number plrŽ B . s max< < : g RŽ G, B .4 . We improve a result of w3x in order to consider the p-local rank of a TI block later. PROPOSITION 2.1. A block B g BlkŽ G . is a TI defect block if and only if either plrŽ B . s 0 or plrŽ B . s 1 and Op Ž G . s 1.
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Proof. Suppose B is a TI defect block with a defect group D s DŽ B .. Then D is a radical subgroup of G and so Op Ž G . F D, where a subgroup R of G is radical if R s Op Ž NG Ž R ... Let G s GrOp Ž G . and denote by x the image of an element x g G in G. Then NG Ž D . s NG Ž D . and x s xOp Ž G . g G _ NG Ž D . for some x g G if and only if x g G _ NG Ž D .. It follows that if D / Op Ž G ., then DrOp Ž G . is a TI set in GrOp Ž G . and by w3, Proposition 5.1x, plrŽ B . s 1. Thus either plrŽ B . s 0 or 1. In the latter case, Op Ž G . s 1 since Op Ž G . F D l D x for each x g G. Suppose plrŽ B . s 0. Since D is radical in G, it follows that D s Op Ž G ., and since D is normal in G, it is a TI set and B is a TI defect block. Suppose plrŽ B . s 1 and Op Ž G . s 1. Then D / 1 and again by w3, Proposition 5.1x, D s DrOp Ž G . is a TI set of G s GrOp Ž G ., so that B is a TI defect block.
3. TI AND TI DEFECT BLOCKS LEMMA 3.1. Let B be a TI block Ž D, bD . a Sylow B-subgroup and Ž Q, bQ . F Ž D, bD .. Then there is a B-subgroup Ž P, bP . with Ž Q, bQ .eŽ P, bP . F Ž D, bD . such that NG Ž Q, bQ . F NG Ž P, bP .. Proof. Let P s ND Ž Q . ) Q. Then there is an unique Ž P, bP . such that
Ž Q, bQ . e Ž P , bP . F Ž D, bD . . Since B is TI, we have NG Ž Q, bQ . F NG Ž D, bD . F NG Ž D .. So NG Ž Q, bQ . F NG Ž P .. Let x g NG Ž Q, bQ .. Then Ž P, bP . x s Ž P, bPx .. But x g NG Ž D, bD ., so both Ž P, bP . and Ž P, bPx . are contained in Ž D, bP .. Hence x g NG Ž P, bP . and we are done. PROPOSITION 3.2. Let B be a TI block, and let Ž Q, bQ . be a non-tri¨ ial B-subgroup. Suppose Q is a subgroup of a defect group D s DŽ B .. Then Ž Q, bQ . is contained in a unique Sylow B-subgroup Ž D, bD .. Proof. It suffices to show the existence of the Sylow B-subgroup Ž D, bD .. Let Q be maximal in D such that there is a B-subgroup Ž Q, bQ . with no Sylow B-subgroup of the form Ž D, bD . containing it. Clearly Q - D. Let Ž D, bXD . be a fixed Sylow B-subgroup. This defines a B-subgroup Ž Q, b 0 . contained in it. Then by Lemma 3.1 there is a B-subgroup Ž P, b1 . with Ž Q, b 0 .eŽ P, b1 . F Ž D, bXD . and NG Ž Q, b 0 . F NG Ž P, b1 .. Now CG Ž Q . Q F NG Ž Q, b 0 . and NG Ž P, b1 . F NG Ž P ., so CG Ž Q . Q F NG Ž P . and C G Ž Q . P F NG Ž P . . Since C G Ž P . F C G Ž Q . and Ž . Ž . CG Ž P . P e ᎏ NG P , it follows that both CG P P and P are normal subgroups of CG Ž Q . P. If H s CG Ž Q . P, then CH Ž Q . s CG Ž Q ., Ž Q, bQ . is a
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subpair of H and bQH is a well-defined block of H. Set BH s bQH. Then BHG s B as bQG s B, and hence BH g BlkŽ H N B .. Let bP be a block of CG Ž P . P covered by BH . By w1, Theorem 15.1x, bPh s BH s bQH and so Ž Q, bQ .eŽ P, bP .. By the maximality of Q, since Q - P there is a Sylow B-subgroup Ž D, bD . containing Ž P, bP ., and so also Ž Q, bQ ., a contradiction. PROPOSITION 3.3. A TI block B is a TI defect block. In particular, plrŽ B . s 0 or 1, and in the latter case Op Ž G . s 1. Proof. Let B be a TI block and suppose D s DŽ B . is not a TI set. Then 1 / D l D x for some x g G _ NG Ž D .. Let Ž D, bD . be a fixed Sylow B-subgroup and let Q s D l D x. Then Ž Q, bQ . F Ž D, bD . for a unique block bQ g BlkŽ CG Ž Q . Q ., so that Ž Q, bQ . is a non-trivial B-subgroup. Since D x is also a defect group of B and Q F D x , it follows by Proposition 3.2 that Ž Q, bQ . - Ž D x , bx . for some Sylow B-subgroup Ž D x , bx ., which is impossible since Ž D, bD . / Ž D x , bx .. Thus B is a TI defect group, and the last assertion follows by Proposition 2.1. Suppose B s B0 Ž G . is the principal block of G. If B is a TI defect block and if a non-trivial B-subgroup Ž R, bR . is contained in Sylow B-subgroups Ž D, bD . and Ž D⬘, bD ⬘ ., then D⬘ s D x for some x g G and R F D l D⬘ s D l D x. Thus x g NG Ž D . and D s D⬘ as D is a TI set and R / 1. By Brauer’s third main theorem, bD s bD ⬘ is the principal block of CG Ž D . D, so that B is a TI block. It follows that if a principal block is a TI defect block, then it is a TI block. But this is not true in general. We describe where examples of TI defect blocks which are not TI may be found and then give a specific example. Suppose that G has TI Sylow p-subgroups and that these Sylow p-subgroups are abelian and have p-rank greater than one. Then G s Op⬘Ž G ..Ž N. A., where N is a nonabelian simple group and A is a finite group such that N. A acts faithfully on N Žsee, for example, w5, 1.2x.. Let P g Syl p Ž G . Žand note that PCG Ž P . s CG Ž P ... Then Op⬘Ž G . F NG Ž P ., and so Op⬘Ž G . F CG Ž P .. Hence N F X s ² P : P g Syl p Ž G .: e yG, and Op⬘Ž G . centralizes N. So G has the form Ž Op⬘Ž G .夹M .. A for some quasisimple group M with TI Sylow p-subgroups, where Ž Op⬘Ž G .夹M . is a central product of Op⬘Ž G . and M over Op⬘Ž G . l M. To produce examples of TI defect blocks which are not TI, it suffices to find non-abelian simple groups M with TI Sylow p-subgroups and outer automorphism groups A of M, such that for a non-trivial subgroup Q - P we have CAŽ Q . / CAŽ P .. Then we may choose Op⬘Ž G . such that CAŽ Q . and CAŽ P . act non-trivially and trivially on Op⬘Ž G ., respectively. Choose 1 , 2 g IrrŽ Op⬘Ž G .. fixed by
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CAŽ P . but conjugate in CAŽ Q ., and consider bP g BlkŽ CG Ž P .. covering 1 Žafter identifying i with the block of defect zero containing it. and its Brauer correspondent bQ in CG Ž Q .. But bQ must also cover 2 , so there must be a block bXP / bP of CG Ž P . covering 2 which is also a Brauer correspondent of bQ Žconsider the conjugate of bP by the element of x g CG Ž Q . taking 1 to 2 .. This would give the counterexample we require. In particular, consider the following subgroup of S10 . Take M s A 6 F S10 permuting 1, . . . , 6 4 and O p ⬘ Ž G . s ²Ž7, 8, 9, 10 .: . Take P s ²Ž1, 2, 3., Ž4, 5, 6.: and Q s ²Ž4, 5, 6.:. Let A s ²Ž1, 2. Ž7, 10. Ž8, 9.:, and define G s Ž M = Op⬘Ž G ... A. Note that A 6 has TI Sylow 3-subgroups, hence so has G, and every 3-block of G is TI defect. Then CG Ž P . s P = Op⬘Ž G . and so has four 3-blocks. CG Ž Q . s Ž P = Op⬘Ž G ... A and has three 3-blocks. Now every 3-block of CG Ž P . has a Brauer correspondent in CG Ž Q ., and so there must be two distinct 3-blocks bP , bXP of CG Ž P . sharing a 3-block bQ of CG Ž Q .. Hence Ž Q, bQ . is contained in distinct subpairs Ž P, bP . and Ž P, bXP ., so bQG has TI defect groups but is not TI.
4. SOME ALTERNATIVE CHARACTERIZATIONS OF TI BLOCKS A B-subgroup Ž Q, bQ . of G is special if it is maximal amongst those NG Ž Q, bQ .-stable subpairs for NG Ž Q, bQ .. ŽRecall that if Ž R, bR . is a subpair for NG Ž Q, bQ . containing Ž Q, bQ ., then it is necessarily a subpair for G and a B-subgroup.. Note that Ž1, B . and the Sylow B-subgroups are special. Note also that if B0 is the principal p-block, then the special B-subgroups correspond to the radical p-subgroups of G. Before proving our main result we give a well-known but useful lemma: LEMMA 4.1. Let Ž Q, bQ . be a B-subpair, where B is a p-block of a finite group G, and let Ž P, bP . be a B-subpair properly containing Ž Q, bQ .. Then Q - NP Ž Q, bQ .. Proof. Write R s NP Ž Q . ) Q. Then there is a B-subpair Ž R, bR . with Ž Q, bQ . F Ž R, bR . F Ž P, bP .. Since Qe R, it follows that Ž Q, bQ .eŽ R, bR ., where by definition Q - R F NG Ž Q, bQ ., and the result follows. PROPOSITION 4.2. Let B be a p-block of a finite group G with Op Ž G . s 1. Then the following are equi¨ alent: Ži. B is a TI block. Žii. NG Ž D, bD . is a strongly B-embedded subgroup for each Sylow B-subgroup Ž D, bD ..
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Žiii. The only non-tri¨ ial special B-subgroups are the Sylow B-subgroups. Proof. Ži. « Žii. is immediate. Žii. « Žiii. Suppose that Ž Q, bQ . is a non-trivial special B-subgroup. Then Ž Q, bQ . F Ž D, bD . for some Sylow B-subgroup Ž D, bD .. Suppose that Q / D. Then R s ND Ž Q, bQ . ) Q by Lemma 4.1. Let bR be the unique p-block of RCG Ž R . such that Ž Q, bQ . - Ž R, bR . F Ž D, bD .. Then CG Ž R . F CG Ž Q . F NG Ž Q, bQ . and R F NG Ž Q, bQ ., so Ž R, bR . is a subpair for NG Ž Q, bQ .. But NG Ž R, bR . F NG Ž D, bD . by our assumption, so Ž R, bR . is NG Ž Q, bQ .-stable, since NG Ž Q, bQ . F NG Ž D, bD .. Hence Ž R, bR . is a NG Ž Q, bQ .-stable subpair for NG Ž Q, bQ . which strictly contains Ž Q, bQ ., contradicting our assumption that Ž Q, bQ . is special. So Q s D and we are done. Žiii. « Ži. Suppose that B is not a TI, but that condition Žiii. holds. Choose Q maximal for which there is some bQ g BlkŽ QCG Ž Q . ¬ B . such that Ž Q, bQ . is contained in two distinct Sylow B-subgroups Ž D, bD . and Ž D 1 , bD .. Then Ž Q, bQ . is nontrivial, a B-subgroup, and not special. 1 Consider a chain
Ž Q, bQ . s Ž Q0 , b0 . F Ž Q1 , b1 . F Ž Q2 , b2 . F
⭈⭈⭈
of B-subgroups, where Ž Q iq1 , biq1 . is a maximal NG Ž Q i , bi .-stable subpair for NG Ž Q i , bi . containing Ž Q i , bi .. Since G is finite eventually there must be an n such that
Ž Q nq j , Q nqj . s Ž Q n , bn . for every j G 0. But then Ž Q n , bn . is special, and so it is a Sylow B-subgroup, say Ž D⬘, bD ⬘ .. Note that by definition NG Ž Q 0 , b 0 . F NG Ž Q1 , b1 . F ⭈⭈⭈ F NG Ž Q n , bn ., so NG Ž Q, bQ . F NG Ž D⬘, bD ⬘ .. Without loss of generality we may take Ž D⬘, bD ⬘ . s Ž D, bD .. Let R s ND 1Ž Q, bQ ., so Q - R F D 1 by Lemma 4.1. Let bR be the unique block of BlkŽ RCG Ž R .. such that Ž R, bR . F Ž D 1 , bD 1 .. We have CG Ž R . F CG Ž Q . F NG Ž Q, bQ . F NG Ž D, bD . , and also R F NG Ž Q, bQ . F NG Ž D, bD ., so Ž R, bR . is a subpair for NG Ž D, bD . and bRNG Ž D, b D . is defined. But Ž bRNG Ž D, b D . . G s bRG s B, so Ž D, bD . is the unique Sylow bRNG Ž D, b D .-subpair and so contains Ž R, bR .. Hence Ž R, bR . is a B-subgroup contained in distinct Sylow B-subgroups Ž D, bD ., Ž D 1 , bD 1 ., contradicting our choice Q. Remarks. Ži. It is easy to see that if B is a TI defect block, then G possesses a strongly B-embedded subgroup Žfor if Ž D, bD . is a Sylow B-subpair for a TI defect block B, and Ž Q, bQ . F Ž D, bD ., then NG Ž Q, bQ .
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F NG Ž Q . F NG Ž D . - G .. However, this does not lead to a strategy for proving that every TI block is TI defect by way of Proposition 4.2, for it is not necessarily the case that if G possesses a strongly B-embedded subgroup then B is a TI. For example, consider the principal 5-block of M22 Žsee w7, 24.2x.. Žii. Special subpairs may be seen as a generalization of radical p-subgroups. Define a special chain of subpairs and the special local rank analogously to the p-local rank, but use special subpairs instead of radical p-subgroups Žso a chain : Ž Q 0 , b 0 . - ⭈⭈⭈ - Ž Q n , bn . of B-subgroups is special if Ž Q 0 , b 0 . is special and Ž Q iq1 , biq1 . is a special subpair of NG Ž i . for each i .. It follows from Propositions 2.1 and 4.2 that if B is a p-block of special local rank one of a finite group with Op Ž G . s 1, then plrŽ B . s 1. 5. THE STRONGLY k-EMBEDDED B-CORE Just as ⌫P, k Ž G . plays a role ˆ in the fusion of p-subgroups Žsee w4x., we demonstrate that ⌫Ž D, b D ., k Ž G . plays a similar part in the fusion of certain B-subgroups: LEMMA 5.1. Suppose that H F G, b g BlkŽ H ¬ B ., and Ž D, bD . is a Sylow b-subgroup of H which is also a B-subgroup of G. If ⌫Ž D, b D ., k Ž G . F H, then Ž D, bD . is a Sylow B-subgroup of G. Proof. Suppose that Ž D, bD . is not a Sylow B-subgroup. Since Ž D, bD . is a B-subgroup, it follows from Lemma 4.1 that there is a B-subgroup Ž P, bP . with Ž D, bD .eŽ P, bP . and D / P, where P is contained in a defect group for B. But PCG Ž P . F PCG Ž D . F NG Ž D, bD . F H, and so Ž P, bP . is a subpair for H and necessarily a b-subgroup, contradicting our choice of Ž D, bD . as a Sylow b-subgroup. PROPOSITION 5.2. Suppose that H F G, b g BlkŽ H ¬ B ., and Ž D, bD . is a B-subgroup which is also a Sylow b-subgroup. Write SB for the poset of B-subgroups, and SB Ž b, k . s Ž Q, bQ . g SB : QCG Ž Q . F H , bQH s b, m p Ž Q . G k 4 . Then ⌫Ž D, b D ., k Ž G . F H if and only if H controls fusion in SB Ž b, k . and NG Ž Q, bQ . F H for e¨ ery Ž Q, bQ . g SB Ž b, k . and e¨ ery B-subgroup Ž Q, bQ . F Ž D, bD .. Proof. Suppose that ⌫Ž D, b D ., k Ž G . F H. Then by Lemma 5.1 Ž D, bD . is a Sylow B-subgroup. Let Ž Q, bQ . g SB Ž b, k ., and suppose that g g G satisfies Ž Q, bQ . g g SB Ž b, k .. We show that g g H, from which it follows immediately that H controls fusion in SB Ž b, k . and NG Ž Q, bQ . F H for
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every Ž Q, bQ . g SB Ž b, k .. Clearly also NG Ž Q, bQ . F H for every Ž Q, bQ . F Ž D, bD .. Since all Sylow b-subgroups are conjugate in H, there are h1 , h 2 g H such that Ž Q, bQ . h1 , Ž Q, bQ . g h 2 F Ž D, bD .. Let x s hy1 1 gh 2 g G. We may now apply w2, 4.10x Žnote that we may use the results of w2x here despite the differing definitions of a subpair, since there is a 1᎐1 correspondence between p-blocks of QCG Ž Q . and p-blocks of CG Ž Q . which respects the Brauer correspondence and conjugacy in G .: there are B-subgroups
Ž R1 , b1 . , . . . , Ž R n , bn . , all contained in Ž D, bD ., and x i g NG Ž R i , bi ., 1 F i F n, such that Ž Q, bQ . h1 F Ž R1 , b1 ., Ž Q, bQ . h1 x 1 ⭈⭈⭈ x i F Ž R iq1 , biq1 ., for i - n, and x s x 1 ⭈⭈⭈ x n . Manifestly m p Ž R i . G m p Ž Q . G k for each i, so each x i g NG Ž R i , bi . F ⌫Ž D, b D ., k Ž G . F H, and so x g H. Hence g g H as required. Now suppose that H controls fusion in SB Ž b, k . and NG Ž Q, bQ . F H for every B-subgroup Ž Q, bQ . F Ž D, bD .. Then Ž Q, bQ . g SB Ž b, k . for all such Ž Q, bQ ., and H controls fusion of B-subgroups contained in Ž D, bD .. Then it follows immediately from w2, 4.24x that NG Ž Q, bQ . s H l NG Ž Q, bQ . for each such Ž Q, bQ ., and so by definition ⌫Ž D, b D ., k Ž G . F H. REFERENCES 1. J. L. Alperin, ‘‘Local Representation Theory,’’ Cambridge Univ. Press, Cambridge, UK, 1986. 2. J. L. Alperin and M. Broue, ´ Local method in block theory, Ann. of Math. 110 Ž1979., 143᎐157. 3. J. An and C. W. Eaton, The p-local rank of a block, J. Group Theory, 3 Ž2000., 369᎐380. 4. M. Aschbacher, ‘‘Finite Group Theory,’’ Cambridge Univ. Press, Cambridge, UK, 1986. 5. H. I. Blau and G. O. Michler, Modular representation theory of finite groups with T.I. Sylow p-subgroups, Trans. Amer. Math. Soc. 319 Ž1990., 417᎐468. 6. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ‘‘An ATLAS of Finite Groups,’’ Clarendon, Oxford, 1985. 7. D. Gorenstein and R. Lyons, On finite groups of characteristic 2-type Mem. Amer. Math. Soc. 276 Ž1982.. 8. J. B. Olsson, On subpairs and modular representation theory, J. Algebra 76 Ž1982., 261᎐279.