#### Vol. 2, No. 8, 2008

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On Oliver's $p$-group conjecture

### David J. Green, László Héthelyi and Markus Lilienthal

Vol. 2 (2008), No. 8, 969–977
##### 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.

##### Keywords
$p$-group, characteristic subgroup, Thompson subgroup, $p$-local finite group, Replacement Theorem
Primary: 20D15