2-subnormal quadratic offenders and Oliver's p-group conjecture

TL;DR

This paper investigates Oliver's conjecture on the containment of the Thompson subgroup within the Oliver subgroup in finite p-groups, proving it holds under specific structural conditions for p ≥ 5.

Contribution

It introduces the concept of 2-subnormal quadratic offenders and proves Oliver's conjecture for groups with certain class or Baumann subgroup properties.

Findings

Oliver's conjecture holds if the quotient group has class ≤ log2(p-2)+1.

Existence of 2-subnormal quadratic offenders in certain p-groups.

Validation of Oliver's conjecture under specific structural conditions.

Abstract

Bob Oliver conjectures that if $p$ is an odd prime and $S$ is a finite $p$-group, then the Oliver subgroup $\X(S)$ contains the Thompson subgroup $J_e(S)$. A positive resolution of this conjecture would give the existence and uniqueness of centric linking systems for fusion systems at odd primes. Using ideas and work of Glauberman, we prove that if $p \geq 5$, $G$ is a finite $p$-group, and $V$ is an elementary abelian $p$-group which is an F-module for $G$, then there exists a quadratic offender which is 2-subnormal (normal in its normal closure) in $G$. We apply this to show that Oliver's conjecture holds provided the quotient $G = S/\X(S)$ has class at most $\log_2(p-2) + 1$, or $p \geq 5$ and $G$ is equal to its own Baumann subgroup.