Further evidence for conjectures in block theory

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