Further evidence for conjectures in block theory

Sambale, Benjamin

doi:10.2140/ant.2013.7.2241

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Further evidence for conjectures in block theory

Benjamin Sambale

Vol. 7 (2013), No. 9, 2241–2273

DOI: 10.2140/ant.2013.7.2241

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 $D_{2^{n}} \times C_{2^{m}}$ , $Q_{2^{n}} \times C_{2^{m}}$ 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.

Keywords

$2$-blocks, bicyclic defect groups, Brauer's $k(B)$-conjecture, Alperin's weight conjecture

Mathematical Subject Classification 2010

Primary: 20C15

Secondary: 20C20

Milestones

Received: 30 September 2012

Revised: 16 October 2012

Accepted: 23 March 2013

Published: 18 December 2013

Authors

Benjamin Sambale
Mathematisches Institut Friedrich-Schiller-Universität D-07737 Jena Germany

Vol. 7, No. 9, 2013