On Oliver’s p-group conjecture

Green, David; Héthelyi, László; Lilienthal, Markus

doi:10.2140/ant.2008.2.969

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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

DOI: 10.2140/ant.2008.2.969

Abstract

Let $S$ be a $p$ -group for an odd prime $p$ . B. Oliver conjectures that a certain characteristic subgroup $X (S)$ always contains the Thompson subgroup $J (S)$ . 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 $S ∕ X (S)$ has nilpotence class at most two.

Keywords

$p$-group, characteristic subgroup, Thompson subgroup, $p$-local finite group, Replacement Theorem

Mathematical Subject Classification 2000

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

Vol. 2, No. 8, 2008