越谷 重夫

<title>Abstract</title> We present a sufficient condition for the <inline-formula> <alternatives> <tex-math>$kG$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline1.png" /> </alternatives> </inline-formula>-Scott module with vertex <inline-formula> <alternatives> <tex-math>$P$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline2.png" /> </alternatives> </inline-formula> to remain indecomposable under the Brauer construction for any subgroup <inline-formula> <alternatives> <tex-math>$Q$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline3.png" /> </alternatives> </inline-formula> of <inline-formula> <alternatives> <tex-math>$P$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline4.png" /> </alternatives> </inline-formula> as <inline-formula> <alternatives> <tex-math>$k[Q\,C_G(Q)]$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline5.png" /> </alternatives> </inline-formula>-module, where <inline-formula> <alternatives> <tex-math>$k$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline6.png" /> </alternatives> </inline-formula> is a field of characteristic <inline-formula> <alternatives> <tex-math>$2$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline7.png" /> </alternatives> </inline-formula>, and <inline-formula> <alternatives> <tex-math>$P$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline8.png" /> </alternatives> </inline-formula> is a semidihedral <inline-formula> <alternatives> <tex-math>$2$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline9.png" /> </alternatives> </inline-formula>-subgroup of a finite group <inline-formula> <alternatives> <tex-math>$G$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline10.png" /> </alternatives> </inline-formula>. This generalizes results for the cases where <inline-formula> <alternatives> <tex-math>$P$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline11.png" /> </alternatives> </inline-formula> is abelian or dihedral. The Brauer indecomposability is defined by R. Kessar, N. Kunugi and N. Mitsuhashi. The motivation of this paper is the fact that the Brauer indecomposability of a <inline-formula> <alternatives> <tex-math>$p$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline12.png" /> </alternatives> </inline-formula>-permutation bimodule (where <inline-formula> <alternatives> <tex-math>$p$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline13.png" /> </alternatives> </inline-formula> is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method due to Broué, Rickard, Linckelmann and Rouquier, that then can possibly be lifted to a splendid derived (splendid Morita) equivalence.

In modular representation theory of finite groups, there is a well-known and important conjecture due to M. Broue. He has conjectured that, for a prime p, if a finite group G has an abelian Sylow p-subgroup P, then the principal p-blocks of G and the normalizer NG.P. of P in G would be derived equivalent. It is shown here that, the Broue's conjecture is true for a prime 3 and for the projective special unitary group G . PSU.3 q2. for a power q of a prime satisfying q12 or 5 (mod 9). In this case such a G has elementary abelian Sylow 3-subgroups of order 9.

Given an odd prime <inline-formula> <alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0027763017000459_inline1" xlink:type="simple" /><tex-math>$p$</tex-math></alternatives> </inline-formula>, we investigate the position of simple modules in the stable Auslander–Reiten quiver of the principal block of a finite group with noncyclic abelian Sylow <inline-formula> <alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0027763017000459_inline2" xlink:type="simple" /><tex-math>$p$</tex-math></alternatives> </inline-formula>-subgroups. In particular, we prove a reduction to finite simple groups. In the case that the characteristic is <inline-formula> <alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0027763017000459_inline3" xlink:type="simple" /><tex-math>$3$</tex-math></alternatives> </inline-formula>, we prove that simple modules in the principal block all lie at the end of their components.

We give a lower bound of the Loewy length of the projective cover of the trivial module for the group algebra kG of a finite group G of Lie type defined over a finite field of odd characteristic p, where k is an arbitrary field of characteristic p. The proof uses Auslander-Reiten theory.

<title>Abstract</title>We verify the inductive Blockwise Alperin Weight (BAW) and the inductive Alperin–McKay (AM) conditions introduced by the second author, for

<title>Abstract</title>We provide a description of the torsion subgroup

<title>Abstract</title>We give a lower bound on the Loewy length of a <italic>p</italic>-block of a finite group in terms of its defect. We then 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 <italic>p</italic>-solvable groups can only admit blocks of Loewy length 4 if <italic>p</italic>=2. However, we find (principal) blocks of simple groups with Loewy length 4 and defect 1 for all <italic>p</italic> ≡ 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.

Let A be the principal 3-block of a finite group G with an abelian Sylow 3-subgroup P. Let C(A) be the Cartan matrix of A. and we denote by rho(C(A)) the unique largest eigenvalue of C(A). The value rho(C(A)) is called the Frobenius-Perron eigenvalue of C(A). We shall prove that rho(C(A)) is a rational number if and only if A and the principal 3-block of N(G)(P) are Morita equivalent. This generalizes earlier Wada&apos;s theorem in 2007, where he proves it only for the case that the order of P is nine, while we prove it for the case that P is an arbitrary finite abelian 3-group. The result presented here uses the classification of finite simple groups. (C) 2010 Elsevier Inc. All rights reserved.

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broue. He conjectures that, for any prime p. if a p-block A of a finite group G has an abelian defect group P. then A and its Brauer corresponding block B of the normaliser N(G)(P) of P in G are derived equivalent (Rickard equivalent). This conjecture is called Broue's abelian defect group conjecture. We prove in this paper that Broue's abelian defect group conjecture is true for a non-principal 3-block A with an elementary abelian defect group P of order 9 of the Harada-Norton simple group HN. It then turns out that Broue's abelian defect group conjecture holds for all primes p and for all p-blocks of the Harada-Norton simple group HN. (C) 2010 Elsevier Inc. All rights reserved.

We shall present a method to get trivial source modules easily just by looking at values of ordinary characters at non-identity p-elements in finite groups instead of doing huge calculation. The method is only for a case where defect groups are cyclic. Nevertheless, it works well at least when we want to prove Broué's abelian defect group conjecture for blocks which have elementary abelian defect groups of order p2. © Springer-Verlag 2009.

In representation theory of finite groups, one of the most important and interesting problems is that, for a p-block A of a finite group G where p is a prime, the numbers k(A) and ℓ(A) of irreducible ordinary and Brauer characters, respectively, of G in A are p-locally determined. We calculate k(A) and ℓ(A) for the cases where A is a full defect p-block of G, namely, a defect group P of A is a Sylow p-subgroup of G and P is a nonabelian metacyclic p-group Mn+1(p) of order pn+1 and exponent pn for n ≥ 2, and where A is not necessarily a full defect p-block but its defect group P = Mn+1(p) is normal in G. The proof is independent of the classification of finite simple groups. © 2010 Birkhäuser Verlag Basel/Switzerland.