Vol. 7, No. 9, 2013

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Further evidence for conjectures in block theory

Benjamin Sambale

Vol. 7 (2013), No. 9, 2241–2273
Abstract

We prove new inequalities concerning Brauer’s $k\left(B\right)$-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 ${D}_{{2}^{n}}\ast {C}_{{2}^{m}}$. 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 $k\left(B\right)$-conjecture and Olsson’s conjecture for the $2$-blocks of defect at most $5$. This completes results from a previous paper. The $k\left(B\right)$-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
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