Donovan's conjecture and extensions by the centralizer of a defect group

TL;DR

This paper investigates Donovan's conjecture for specific block types in group theory, extending previous results and confirming the conjecture for certain defect groups over discrete valuation rings.

Contribution

It extends Donovan's conjecture results to blocks with non-abelian defect groups and verifies the conjecture for groups involving quaternion and cyclic components.

Findings

Donovan's conjecture holds for blocks with defect groups Q_8 × C_{2^n}

Donovan's conjecture holds for blocks with defect groups Q_8 × Q_8

Results generalize to blocks with certain normal subgroup conditions

Abstract

We consider Donovan's conjecture in the context of blocks of groups $G$ with defect group $D$ and normal subgroups $N \lhd G$ such that $G=C_D(D\cap N)N$, extending similar results for blocks with abelian defect groups. As an application we show that Donovan's conjecture holds for blocks with defect groups of the form $Q_8 \times C_{2^n}$ or $Q_8 \times Q_8$ defined over a discrete valuation ring.