We prove new inequalities concerning Brauer’s -conjecture and Olsson’s conjecture by generalizing old results. After that, we obtain the invariants for -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 , and . 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 -conjecture and Olsson’s conjecture for the -blocks of defect at most . This completes results from a previous paper. The -conjecture is also verified for defect groups with a cyclic subgroup of index at most . Finally, we consider Olsson’s conjecture for certain -blocks.
Benjamin Sambale. "Further evidence for conjectures in block theory." Algebra Number Theory 7 (9) 2241 - 2273, 2013. https://doi.org/10.2140/ant.2013.7.2241