On the reduction of Alperin's Conjecture to the quasi-simple groups

TL;DR

This paper reduces the proof of Alperin's Conjecture to verifying a specific refinement on certain central extensions of groups containing a simple normal subgroup, simplifying the overall proof strategy.

Contribution

It refines previous results by showing the conjecture can be proved by checking a condition on central extensions with simple normal subgroups, improving and repairing earlier arguments.

Findings

Reduction of Alperin's Conjecture verification to central extensions

Verification of the refinement on groups with simple normal subgroups

Correction of previous flawed arguments in earlier work

Abstract

We show that the refinement of Alperin's Conjecture proposed in "Frobenius Categories versus Brauer Blocks", Progress in Math. 274, can be proved by checking that this refinement holds on any central k*-extension of a finite group H containing a normal simple group S with trivial centralizer in H and p'-cyclic quotient H/S. This paper improves our result in [ibidem, Theorem 16.45] and repairs some bad arguments there.