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

Abstract

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

The 2-blocks of defect 4

“…We may therefore clearly assume that N = 1. In this case B is the principal 2-block of PSL (2,16), and l(B) = 15, a final contradiction.…”

“…Since |G : M | is odd, this implies that t = 1. Hence M/N is a simple group with Sylow 2-subgroup D. By Walter's Theorem (see [44]), we must have M/N = PSL (2,16). Note also that M = F * (G).…”

“…For |D| ≤ 8 the block invariants and conjectures for B are known by the work of Brauer [5], Olsson [23] and Kessar, Koshitani and Linckelmann [16]. So we assume that D has order 16.…”

“…The aim of this paper is to show that most of these conjectures are fulfilled if D is small. More precisely, we will assume that B has defect at most 4, i.e., D has order at most 16. We denote the number of irreducible ordinary characters of height i in B by k i (B) for i ≥ 0.…”

Untitled

“…and C 2 n+1 × C 2 for which the block invariants can be calculated by [Usami 1988;Kessar et al 2012]. So let ω = 0.…”