On Oliver's $p$-group conjecture

Abstract

Let $S$ be a $p$-group for an odd prime $p$. B. Oliver conjectures that a certain characteristic subgroup $\mathfrak{X}\left(S\right)$ always contains the Thompson subgroup $J\left(S\right)$. We obtain a reformulation of the conjecture as a statement about modular representations of $p$-groups. Using this we verify Oliver’s conjecture for groups where $S∕\mathfrak{X}\left(S\right)$ has nilpotence class at most two.