A note on 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 odd prime p-groups, providing a key equivalence and an application related to subgroup indices.

Contribution

It establishes an if-and-only-if condition linking the containment of J(S) in al(S) and al_1(S), and proves that the index of J(S)al(S) over al(S) is never equal to p.

Findings

J(S) al(S) ext{ if and only if } J(S) \u001cal_1(S)

The index al(S) of J(S)al(S) over al(S) is not equal to p

Provides a criterion for the containment of Thompson subgroup in Oliver subgroup

Abstract

Let $S$ be a $p$-group for an odd prime $p$, Oliver proposed the conjecture that the Thompson subgroup $J(S)$ is always contained in the Oliver subgroup $\mathfrak{X}(S)$. That means he conjectured that $|J(S)\mathfrak{X}(S):\mathfrak{X}(S)|=1$. Let $\mathfrak{X}_1(S)$ be a subgroup of $S$ such that $\mathfrak{X}_1(S)/\mathfrak{X}(S)$ is the center of $S/\mathfrak{X}(S)$. In this short note, we prove that $J(S)\leq \mathfrak{X}(S)$ if and only if $J(S)\leq \mathfrak{X}_1(S)$. As an easy application, we prove that $|J(S)\mathfrak{X}(S):\mathfrak{X}(S)|\neq p$.