Equivariant Alperin-Robinson's Conjecture reduces to almost-simple k*-groups

TL;DR

This paper reduces the verification of the equivariant Alperin-Robinson's Conjecture to checking it on central extensions of finite almost-simple or simple groups, simplifying the proof process using classification results.

Contribution

It demonstrates that the equivariant Alperin-Robinson's Conjecture can be verified by examining central k*-extensions of almost-simple or simple groups, streamlining previous approaches.

Findings

Reduction of the conjecture to almost-simple groups

Verification on central k*-extensions suffices

Appendix develops supporting arguments

Abstract

In a recent paper, Gabriel Navarro and Pham Huu Tiep show that the so-called Alperin Weight Conjecture can be verified via the Classification of the Finite Simple Groups, provided any simple group fulfills a very precise list of conditions. Our purpose here is to show that the equivariant refinement of the Alperin's Conjecture for blocks formulated by Geoffrey Robinson in the eighties can be reduced to checking the same statement on any central k*-extension of any finite almost-simple group, or of any finite simple group up to verifying an "almost necessary" condition. In an Appendix we develop some old arguments that we need in the proof.