On Oliver's p-group conjecture

TL;DR

This paper reformulates Oliver's p-group conjecture in terms of modular representations and verifies it for groups where the quotient S/X(S) has nilpotence class at most two.

Contribution

The paper provides a new reformulation of Oliver's conjecture using modular representation theory and proves it for a specific class of p-groups.

Findings

Verification of Oliver's conjecture for groups with S/X(S) of nilpotence class ≤ 2

Reformulation of the conjecture in terms of modular representations

Insight into the structure of characteristic subgroups in p-groups

Abstract

Let S be a p-group for an odd prime p. Bob Oliver conjectures that a certain characteristic subgroup X(S) always contains the Thompson subgroup J(S). We obtain a reformulation of the conjecture as a statement about modular representations of p-groups. Using this we verify Oliver's conjecture for groups where S/X(S) has nilpotence class at most two.