Further evidence for conjectures in block theory

Abstract

We prove new inequalities concerning Brauer's k(B)-conjecture and Olsson's conjecture by generalizing old results. After that, we obtain the invariants for 2-blocks of finite groups with certain bicyclic defect groups. Here, a bicyclic group is a product of two cyclic subgroups. This provides an application for the classification of the corresponding fusion systems in a previous paper. To some extent, this generalizes previously known cases with defect groups of types D 2 n × C 2 m , Q 2 n × C 2 m and D 2 n * C 2 m. As a consequence, we prove Alperin's weight conjecture and other conjectures for several new infinite families of nonnilpotent blocks. We also prove Brauer's k(B)-conjecture and Olsson's conjecture for the 2-blocks of defect at most 5. This completes results from a previous paper. The k(B)-conjecture is also verified for defect groups with a cyclic subgroup of index at most 4. Finally, we consider Olsson's conjecture for certain 3-blocks.

Blocks of Finite Groups and Their Invariants

“…This result from [252] generalizes an old theorem by Olsson [216] for the case u D 1. For the prime p D 2 we also prove Brauer's k.B/-Conjecture under the weaker hypothesis l.b u / Ä 3.…”

“…The results in this section were taken from [252]. It is obvious that Theorem 4.2 should be stronger for small values of l.b u /.…”

“…Later we give corollaries for blocks with bicyclic defect groups. The material comes from [252,257]. We will use the following notation: A group P is called minimal non-abelian of type .r; s/ if P Š hx; y j x p r D y p s D OEx; y p D OEx; x; y D OEy; x; y D 1i for r s 1 (see Chap.…”

“…In this chapter which is taken from [114,252] we will use Theorem 4.12 to show that Olsson's Conjecture is satisfied for controlled blocks B with certain defect groups D. Recall that in this situation all subgroups of D are fully F -normalized where F is the fusion system of B. In particular for a subsection .u; b u / the block b u has defect group C D .u/ (cf.…”

“…In the following we consider these cases in detail. The results come from [252]. / has maximal class).…”