Fusion systems on bicyclic 2-groups

TL;DR

This paper classifies all saturated fusion systems on bicyclic 2-groups and applies the results to prove Olsson's Conjecture for blocks with such defect groups.

Contribution

It extends classification of fusion systems to bicyclic 2-groups and confirms Olsson's Conjecture in this context.

Findings

Complete classification of fusion systems on bicyclic 2-groups

Proof of Olsson's Conjecture for blocks with bicyclic defect groups

Extension of previous work on metacyclic 2-groups

Abstract

We classify all (saturated) fusion systems on bicyclic 2-groups. Here, a bicyclic group is a product of two cyclic subgroups. This extends previous work on fusion systems on metacyclic 2-groups (see [Craven-Glesser, 2012] and [Sambale, 2012]). As an application we prove Olsson's Conjecture for all blocks with bicyclic defect groups.