On the Principal Blocks of Finite Groups with Abelian Sylow p-Subgroups

✍ Scribed by Masato Sawabe; Atumi Watanabe

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 149 KB
• Volume: 237
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2000.8460

✦ Synopsis

Let p be a prime number and K be an algebraically closed field of characteristic p. Let G be a finite group and B be a (p-) block of G. We denote by l B the number of isomorphism classes of irreducible KG-modules in B. Let D be a defect group of B and let B 0 be the Brauer correspondent of B, that is, B 0 is a block of N G D associated with B in the sense of Brauer. In [1], Alperin defined a weight for G and raised a conjecture on l B in terms of weights. A weight for G is a pair Q S where Q is p-subgroup of G and S is an irreducible KN G Q -module which is projective when we regard it as an KN G Q /Q-module. When S belongs to a block of N G Q associated with B, Q S is called a B-weight. Alperin conjectured that l B is equal to the number of G-conjugacy classes of B-weights. Suppose that D is abelian for a while. Then any B-weight is G-conjugate to D V where V is an irreducible KN G D -module belonging to B 0 (see [1], p. 372). So it is conjectured that l B = l B 0 . This is well known to be true for B when D is cyclic, by Dade's result on blocks with cyclic defect groups. It is true for principal 2-blocks too by [9]. Let b be a block of C G D associated with B. When N G D b /C G D is an elementary abelian 2-group or when N G D b /C G D is small, Alperin's conjecture is true for B (see [14,15,[17][18][19][20][21]23] for the details).