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

Charles W Eaton,Michael Livesey

doi:10.1016/j.jalgebra.2021.04.029

Abstract

We consider Donovan's conjecture in the context of blocks of groups G with defect group D and normal subgroups N◁G such that G=CD(D∩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 Q8×C2n or Q8×Q8 defined over a discrete valuation ring.

Highlights

Donovan’s conjecture states that for a given finite p-group P, there are only finitely many Morita equivalence classes amongst blocks of finite groups with defect groups isomorphic to P
We show that Donovan’s conjecture with respect to O holds when D ∼= Q8 × C2n or Q8 × Q8 for some n
In order to verify Donovan’s conjecture for P for O-blocks, it suffices to check it for blocks of finite groups G with defect group D isomorphic to a subgroup of P and no proper normal subgroup N ✁ G such that G = CD(D ∩ N )N

Summary

Introduction

In [21] Kessar showed that the k-Donovan conjecture is equivalent to showing Conjecture 1.1 and that the Morita Frobenius number (defined in §2) of a block is bounded in terms of the order of the defect groups. Our first reduction result concerns the other half of the problem, i.e., bounding the strong O-Frobenius numbers in terms of the defect groups: Theorem 1.2 Let G be a finite group and B be a block of OG with defect group D. In order to verify Donovan’s conjecture for P for O-blocks, it suffices to check it for blocks of finite groups G with defect group D isomorphic to a subgroup of P and no proper normal subgroup N ✁ G such that G = CD(D ∩ N )N.

Strong Frobenius numbers and extensions by the centralizer of a defect group

A reduction theorem for Donovan’s conjecture

Blocks of quasisimple groups with defect groups