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

Abstract

We show that the refinement of Alperin's Conjecture proposed in L. Puig (2009) [10, Chap. 16] 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 L. Puig (2009) [10, Theorem 16.45] and repairs some bad arguments there.

Local-global conjectures in the representation theory of finite groups

“…In the same paper, they verified their inductive AWC condition for example for groups of Lie type in their defining characteristic, as well as for all simple groups with abelian Sylow 2-subgroups, while An and Dietrich [6] show it for sporadic groups. This reduction was then refined by Späth [94] to treat the block-wise version: Puig [83,84] has announced another reduction of Conjecture 4.1 to nearly simple groups.…”

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

“…Introduction 1.1. In a recent paper [3], 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 that they consider easier to check than ours, firstly stated in [6,Theorem 16.45] and significantly weakened in [8,Theorem 1.6] † †. As a matter of fact, our reduction result concerns Alperin's Conjecture for blocks in an equivariant formulation which goes back to Geoffrey Robinson in the eighties (it appears in his joint work [11] with Reiner Staszewski).…”

“…1.2. Actually, in the introduction of [6] -from I29 to I37 -we consider a refinement of Alperin-Robinson's Conjecture for blocks; but, only in [8] we really show that its verification can be reduced to check that the same refinement holds on the so-called almost-simple k * -groups -namely, central k * -extensions of finite groups H containing a normal simple subgroup S such that H/S is a cyclic p ′ -group and we have C H (S) = {1} . To carry out this checking obviously depends on admitting the Classification of the Finite Simple Groups, and our proof of the reduction itself uses the solvability of the outer automorphism group of a finite simple group (SOFSG), a known fact whose actual proof depends on this classification.…”

A reduction theorem for the blockwise Alperin weight conjecture

“…There are some general results towards proving the conjecture: In [18], Knörr and Robinson gave a reformulation of this conjecture using alternating sums and descending chains of p-subgroups of G. Puig has refined this conjecture and, in [29,30], given a theorem reducing it to a statement about non-abelian groups, closely related to simple groups.…”

“…While this paper was being finalized, Puig, in [31,32], published a result similar to Theorem A depending on methods and results from [29,30]. Our approach is completely independent.…”