#### 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
##### Milestones
Received: 17 April 2008
Revised: 14 August 2008
Accepted: 19 September 2008
Published: 23 November 2008
##### Authors
 David J. Green Mathematical Institute Friedrich-Schiller-Universität Jena 07737 Jena Germany http://users.minet.uni-jena.de/~green/index-en.php László Héthelyi Department of Algebra Budapest University of Technology and Economics Budapest, Pf. 91 H-1521 Hungary Markus Lilienthal FB Wirtschaftswissenschaften Johann-Wolfgang-Goethe-Universität House of Finance Grüneburgplatz 1 60325 Frankfurt (Main) Germany