A note on Oliver's p-group conjecture

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 $|J(S)\mathfrak{X}(S):\mathfrak{X}(S)|\neq p$.