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 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 SX(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