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