Shigeo Koshitani

Conjectures of Alperin and Broué for 2-blocks with elementary abelian defect groups of order 8

Abstract Using the classification of finite simple groups, we prove Alperins weight conjecture and the character theoretic version of Broués abelian defect conjecture for 2-blocks of finite groups with an elementary abelian defect group of order 8.

On Loewy lengths of blocks

We give a lower bound on the Loewy length of a p-block of a finite group in terms of its defect. Afterwards we discuss blocks with small Loewy length. Since blocks with Loewy length at most 3 are known, we focus on blocks of Loewy length 4 and provide a relatively short list of possible defect groups. It turns out that p-solvable groups can only admit blocks of Loewy length 4 if p = 2. However, we find (principal) blocks of simple groups with Loewy length 4 and defect 1 for all p ≡ 1 (mod 3). We also consider sporadic, symmetric and simple groups of Lie type in defining characteristic. Finally, we give stronger conditions on the Loewy length of a block with cyclic defect group in terms of its Brauer tree.

DONOVAN CONJECTURE AND LOEWY LENGTH FOR PRINCIPAL 3-BLOCKS OF FINITE GROUPS WITH ELEMENTARY ABELIAN SYLOW 3-SUBGROUP OF ORDER 9

In modular representation theory of finite groups there has been a well-known conjecture due to P. Donovan. Donovan conjecture is on blocks of group algebras of finite groups over an algebraically closed field k of prime characteristic p, which says that, for any given finite p-group P, up to Morita equivalence, there are only finitely many block algebras with defect group P. We prove that Donovan conjecture holds for principal block algebras in the case where P is elementary abelian 3-group of order 9. Moreover, under the same assumption, namely, if G is a finite group with elementary abelian Sylow 3-subgroup P of order 9, then the Loewy length of the principal block algebra B 0 (kG) of the group algebra kG is 5 or 7. The results here depend on the classification of finite simple groups.

A SPLITTING THEOREM FOR BLOCKS

Let F be an algebraically closed field of prime characteristic p, let G be a finite group, and let H be a normal subgroup of G such that G/H is a ^-group. Moreover, let B be a block of the group algebra FH of H over F. By Osimas theorem, there is a unique block A of FG covering B. We are interested in the structure of A. As usual, the general case reduces to the special one where B is (/-stable. Thus we assume in the following that B is G-stable and denote by P a defect group of A. Then Q\—Pr\H is a defect group of B, and G = F//(see [3, V]). If P is abelian then the character theory of A is described in a paper by R. Knόrr [5]. We are interested in the structure of A as a ring under the additional hypothesis that Q has a complement in P. We prove that such a splitting of defect groups implies a splitting of blocks:

The inductive Alperin–McKay and Blockwise Alperin Weightconditions for blocks with cyclic defect groups and odd primes

Abstract We verify the inductive Blockwise Alperin Weight (BAW) and the inductive Alperin–McKay (AM) conditions introduced by the second author, for p-blocks of finite quasisimple groups with cyclic defect groups whenever the prime p is odd. Furthermore, we establish a criterion that describes conditions under which the inductive AM condition for blocks with abelian defect groups implies the inductive BAW condition for those blocks.

On the Jacobson radical of a block ldeal in a finite p-solvable group for p ⩾ 5

Let F be a field of characteristic p > 0, G a finite p-solvable group, P a Sylow p-subgroup of G of order pr, FG the g;oup algebra of G over F, and J(FG) the Jacobson radical of FG. Following Wallace [ 191 we write t(G) for the least integer t > 1 such that J(FG)’ = 0. We are interested in relations between t(G) and the structure of G. Since J(EG) = E @,J(FG) for any extension field E of F (cf. [ 12, Proposition 12.1 I I), we may assume that F is algebraically closed. Tsushima [ 161 showed t(G) 2 the following are equivalent:

Conjectures of Donovan and Puig for Principal 3-Blocks with Abelian Defect Groups

Abstract One of the most important and interesting conjectures in representation theory of finite groups is Donovans conjecture. It is on block algebras of group algebras of finite groups over an algebraically closed field of prime characteristic p. Donovan conjectures that, for any given finite p-group D, up to Morita equivalence, there are only finitely many block algebras with defect group D. We prove in this article that Donovans conjecture is true for principal block algebras in the case where D is an arbitrary finite abelian 3-group. There is another important and interesting conjecture due to Puig, which is stated as the above conjecture if we replace “Morita equivalence” by “Puig equivalence”. We prove in this paper that Puigs conjecture is true for principal 3-block algebras of finite groups with an elementary abelian Sylow 3-subgroup of order 9. Actually, we prove even more. It is proven in this paper also that there are exactly 22 non-Puig equivalent classes of principal block algebras of finite groups over a complete discrete valuation ring of rank one if a Sylow 3- subgroup D of them is an elementary abelian 3-group of order 9. The results here depend on the classification of finite simple groups.

Cartan invariants of group algebras of finite groups

We give a result on Cartan invariants of the group algebra kG of a finite group G over an algebraically closed field k, which implies that if the Loewy length (socle length) of the projective indecomposable kG-module corresponding to the trivial kG-module is four, then k has characteristic 2. The proof is independent of the classification of finite simple groups. 0. INTRODUCTION AND NOTATION Let kG be the group algebra of a finite group G over a field k of characteristic p > 0. By a kG-module we always mean a right kG-module. In this paper we discuss Cartan invariants of kG, especially those of the projective cover P = P(kG) of the trivial kG-module kG. Let j be the Loewy length of P, that is, j is the least positive integer t such that pJt = 0 where J is the Jacobson radical of kG. It is well-known that the structure of G is completely determined provided j 0. Assume that S is a simple kG-module which is self-dual, that is, the dual module S* = Homk(S, k) of S is isomorphic to S itself as kG-modules. If p is odd, then there is a simple kG-module T such that T is self-dual and the Cartan invariant c(S, T) with respect to S and T is odd. Corollary. Let k be an arbitrary field and G a finite group. If the Loewy length of the projective indecomposable kG-module corresponding to the trivial kG-module is four, then k has characterstic 2. Received by the editors August 17, 1994 and, in revised form, February 7, 1995. 1991 Mathematics Subject Classification. Primary 20C05, 20C20. Supported in part by the Alexander von Humboldt Foundation, the Mathematical Prizes Fund, the University of Oxford and the Sasakawa Foundation. ( 1996 American Mathematical Society 2319 This content downloaded from 157.55.39.215 on Tue, 30 Aug 2016 04:58:29 UTC All use subject to http://about.jstor.org/terms 2320 SHIGEO KOSHITANI Note that the theorem and the corollary do not depend on the classification of finite simple groups. Remark. It should be remarked that if G is the symmetric group on 4 letters and if k has characteristic 2, then the Loewy length of the projective indecomposable kGmodule corresponding to the trivial kG-module is four. The situation for p-solvable groups is treated completely in our previous paper [2]. Throughout this paper we use the following notation and terminology. By a kG-module we mean a finitely generated right kG-module. We write kG for the trivial kG-module, and P(kG) for its projective cover. We denote by J(kG) the Jacobson radical of kG. Let M be a kG-module. Then j(M) denotes the Loewy length of M, namely, j(M) is the least positive integer j such that M.J(kG)i = 0, and Soc(M) denotes the socle of M (see [3, I ?8]). We write M* for the dual module of M, that is, M* = Homk(M, k) and this is also a right kG-module (see [3, I ?6]). Then, M is called self-dual if M M* as kG-modules. For kG-modules M and N, we denote diMk [HOmkG (M, N)] by [M, N]G. For simple kG-modules S and T, c(S, T) denotes the Cartan invariant with respect to S and T (usually, a notation CS,T is used instead). Other notation and terminology follow the books of Feit [1] and Landrock [3]. 1. CALCULATION OF DETERMINANTS In this section we give several lemmas on elementary computation of determinants for matrices over a field of characteristic 2. Throughout this section all entries of matrices are elements in a field of characteristic 2. This assumption is essential here. Lemma 1.1. For non-negative integers m and n, let A be a square matrix of size m + 2n + 2 of the form 1 ... m 1 1* ... n n* u v 1 / aii a ... an an 0 0 m am, am, *** amn amn 0 0 A ?0 bo1 ... bom Col do, Con don uo VO 0* bol ... bOm do, Co1 don COn Uo Uo n bnl ... bnm Cn 1 dn1 Cnn dnn un Vn n bnl ... bnm dn1 Cni ... dnn Cnn Un Vn where S is an (m x m) -matrix. Then, det A = 0 . Proof. We prove the lemma by induction on n. If n = 0, then det A = 0 since two rows indexed by 0 and 0* are the same. Assume n ? 1. Expand A with respect to the last column indexed by v. Then, by symmetry, it is enough to show that the This content downloaded from 157.55.39.215 on Tue, 30 Aug 2016 04:58:29 UTC All use subject to http://about.jstor.org/terms CARTAN INVARIANTS 2321 determinant of the following square matrix B of size m + 2n + 1 is zero: 1 ... .. m 1 1* ... n n* u 1 z all all aln aln 0 : L} ~ : : : : : : m amni aml amn amn 0 0 + 0* 0 ... 0 eol eol ... eon eOn 0 1 bil ... bim Cii di, Cln di, Ul 1* bil ... bim di, Cii ... dln Cln Ul n bnl ... bnm Cnl dnl Cnn dnn Un n* bnl ... bnm dnl Cnl ... dnn Cnn Unr where eoi = coi + doi for i = 1,..., n. Now, expand B with respect to the final column indexed by u. So, again by symmetry, it suffices to prove det C = 0 for the following square matrix C of size m + 2n: 1 ... m 1 1* ... n n* 1 z all all aln aln S m aml ami amn amn 0 + 0* 0 ... 0 eol eol ... eOn eOn C 1 + 1* 0 ... 0 ell ell ... eln eln 2 b2l ... b2m C21 d21 C2n d2n 2* b2l ... b2m d2l C21 ... d2n C2n n bnl ... bnm Cnl dnl Cnn dnn n* bnl ... bnm dnl Cnl .. dnn Cnn where eli = cli + dii for i = 1, ..., n. Then, by carrying two rows indexed by 0 + 0* and 1 + 1* to the end, and then by taking its transpose, we have det C = 0 by induction. O Lemma 1.2. For non-negative integers m and n, let A be a square matrix of size m + 2n + 1 of the form u 1 ... m 1 1* ... n n* V 0 0 ... 0 Vi Vi ... Vn Vn,V 1 0 all all aln aln : : S : : : :. m 0 ami ami amn amn A = 1 ul bil ... bim Cii di1 Cln dln 1* Ul bil ... bim di, Cii *.. dln Cln n Un bnl ... bnm Cnl dnl Cnn dnn n* Un bnl ... bnm dnl Cnl ... dnn Cnn where S is an (mxm)-matrix. Then, det A = 0. This content downloaded from 157.55.39.215 on Tue, 30 Aug 2016 04:58:29 UTC All use subject to http://about.jstor.org/terms 2322 SHIGEO KOSHITANI Proof. First of all, expand A with respect to the first row indexed by v, and sum each pair of determinants of size m + 2n that have the same coefficient vi for = 1,...,n. Namely, we can write detA = Ein4videtBi, where each Bi is a square matrix of size m + 2n and each Bi has the same form as in Lemma 1.1 by a suitable exchanging of columns. Therefore, det A = 0 by Lemma 1.1. O 2. LEMMAS In this section we state several lemmas which will be used in the proofs of our results. Throughout this section we assume that k is an algebraically closed field of characteristic p > 0, and we fix a finite group G such that p divides the order of G. Lemma 2.1 (Webb [9, Theorem E). Let P = P(kG) and assume j(P) ?3 3. If p is odd, then P.J(kG)/Soc(P) is an indecomposable kG-module. Lemma 2.2 ([6, Lemma 1.2]). If M is an indecomposable kG-module with j(M) = 2, then M.J(kG) = Soc(M). Lemma 2.3 ([3, II Corollary 6.9]). For kG-modules M and N, [M,N]G = [N*,M*]G. Lemma 2.4 ([3, I Lemma 8.4 (i)]). For a kG-module M, [M/(M.J(kG))]* Soc(M*) as kG-modules. Lemma 2.5 (Landrock). For simple kG-modules S and T, c(S, T) = c(T, S) = c(S*, T*) = c(T*, S*). Proof. We get the assertion from [1, I Lemma 14.9 and Theorem 16.7] and Landrocks result [4, Theorem A] (cf. [3, I Theorem 9.9]). O 3. PROOFS In this section we give proofs of the theorem and the corollary in the introduction. Proof of Theorem. Let B be a block ideal of kG containing S. Since S is self-dual, Y* is a simple kG-module in B again if Y is a simple kG-module in B (see [3, I Proposition 10.8]). Thus, let So = S,Sj,...,Sm,Ti,Tj*, ..., Tn Tn* all be nonisomorphic simple kG-modules in B; and Si Si* for all i and Tj Tj * for all j. We denote by C the Cartan matrix for B, and let C be its image induced by the canonical epimorphism Z -> 2/2Z. Now, suppose that c(So, Si) is even for all i = 0, ..., m. Then Lemma 2.5 implies that C has the same form as in Lemma 1.2, so that det C = 0 from Lemma 1.2. This means det C is even, which contradicts Brauers result [1, IV Theorem 3.9]). OI Proof of Corollary. First of all, we may assume that k is algebraically closed (see [5, Proposition 12.11]). Let J = J(kG), P = P(kG) and M = PJ/Soc(P). Assume p is odd. By Lemmas 2.1 and 2.2, M is an indecomposable kG-module with MJ = Soc(M). Then, the Theorem implies that there is a simple kG-module T such that T is self-dual and c(kG, T) is odd. On the other hand, since T and M are both self-dual, and since MJ = Soc(M), it follows from Lemmas 2.3 and 2.4 that [M/MJ,T]G = [T,(M/MJ)*]G = [T, Soc(M)]G = [T, MJ]G, This content downloaded from 157.55.39.215 on Tue, 30 Aug 2016 04:58:29 UTC All use subject to http://about.jstor.org/terms CARTAN INVARIANTS 2323 which says that the multiplicities of T in M/MJ and MJ as direct summands are the same, so that c(kG, T) is even, a contradiction. O ACKNOWLEDGEMENTS The author is so grateful to the referee for her or his nice and kind advice, which has improved the main result in the first version of the paper. A part of this work was done while the author was staying at the Institute for Experimental Mathematics, Essen University (July October 1992), and at the Mathematical Institute, University of Oxford (September 1992). He would like to thank the Humboldt Foundation, the Mathematical Prizes Fund, the University of Oxford and the Sasakawa Foundation for their financial support, and also he would like to express his great thanks to Professor G.O. Michler and the people around him, and Professor K. Erdmann for their hospitality.

A remark on the Jacobson radical of a block ideal in a finite p-solvable group

107 We use the notation and terminology as in our previous paper [9 1. Throughout this paper we fix a field F of characteristic p > 0 and a finite p- solvable group G. All groups considered here are finite. Further notation and terminology follow the books of Dornhoff Cl j and Gorenstein [S]. The author would like to express his great thanks to otose who kindly showed the author his preprint [ 1 I]. First of all, we state a generalization of our previous result [9? Lemma 71, which is obtained by making use of otose’s recent result [ll]. Here, we use the notation M(p) and M,(p) as in [5, p. 203 and p. 190]. that is to say, M(p)=(a,b,cl lal=Jbl=Ic~=p,a-‘ba=bc,a-lca=c,b-lcb=c) for p 3 3, M,.(p) = (a, b 1 /aI = p, Ibl = pr- I, a-‘ba = b”) where rn=~‘-~ + 1, for r34 ifp=2; and for ~33 ifp33.

The Projective Cover of the Trivial Module Over a Group Algebra of a Finite Group

We determine all finite groups G such that the Loewy length (socle length) of the projective cover P(k G ) of the trivial kG-module k G is four, where k is a field of characteristic p > 0 and kG is the group algebra of G over k, by using previous results and also the classification of finite simple groups. As a by-product we prove also that if p = 2 then all finite groups G such that the Loewy lengths of the principal block algebras of kG are four, are determined.