Weak closure and Oliver's p-group conjecture

TL;DR

This paper constructs a specific p-group that refutes the weakly closed conjecture but still satisfies Oliver's p-group conjecture, challenging previous verification methods.

Contribution

It provides a counterexample of order 3^49 that separates the weakly closed conjecture from Oliver's conjecture.

Findings

Counterexample of order 3^49 refutes the weakly closed conjecture

The constructed group satisfies Oliver's p-group conjecture

Challenges the approach of verifying Oliver's conjecture via the weakly closed conjecture

Abstract

To date almost all verifications of Oliver's p-group conjecture have proceeded by verifying a stronger conjecture about weakly closed quadratic subgroups. We construct a group of order 3^n for n = 49 which refutes the weakly closed conjecture but satisfies Oliver's conjecture.