A note on Oliver's p-group conjecture

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 $\mathrm{\mid }J(S)\mathfrak{X}(S):\mathfrak{X}(S)\mathrm{\mid }=1$. Let ${\mathfrak{X}}_1(S)$ be a subgroup of $S$ such that ${\mathfrak{X}}_1(S)\mathrm{/}\mathfrak{X}(S)$ is the center of $S\mathrm{/}\mathfrak{X}(S)$. In this short note, we prove that $J(S)\leq \mathfrak{X}(S)$ifandonlyif$J(S)\leq \mathfrak{X}_1(S)$. As an easy application, we prove that $\mathrm{\mid }J(S)\mathfrak{X}(S):\mathfrak{X}(S)\mathrm{\mid }\mathrm{\ne }p$.