On Oliver's p-group conjecture: II

TL;DR

This paper investigates Oliver's p-group conjecture, focusing on the inclusion of the Thompson subgroup within a characteristic subgroup, and verifies the conjecture for many groups via representation theory.

Contribution

It extends previous work by verifying Oliver's conjecture for a broad class of p-groups using representation-theoretic methods.

Findings

Oliver's conjecture holds for many p-groups S/X(S).

The approach links the conjecture to the representation theory of factor groups.

Verification covers a wide variety of group structures.

Abstract

Let p be an odd prime and S a finite p-group. B. Oliver's conjecture arises from an open problem in the theory of p-local finite groups. It is the claim that a certain characteristic subgroup X(S) of S always contains the Thompson subgroup. In previous work the first two authors and M. Lilienthal recast Oliver's conjecture as a statement about the representation theory of the factor group S/X(S). We now verify the conjecture for a wide variety of groups S/X(S).