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2-subnormal quadratic offenders and Oliver's p-group conjecture

Part of: Representation theory of groups

Published online by Cambridge University Press:  10 December 2012

Justin Lynd
Justin Lynd*
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA
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Abstract

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Bob Oliver conjectures that if p is an odd prime and S is a finite p-group, then the Oliver subgroup contains the Thompson subgroup Je(S). A positive resolution of this conjecture would give the existence and uniqueness of centric linking systems for fusion systems at odd primes. Using the ideas and work of Glauberman, we prove that if p ≥ 5, G is a finite p-group, and V is an elementary abelian p-group which is an F-module for G, then there exists a quadratic offender which is 2-subnormal (normal in its normal closure) in G. We apply this to show that Oliver's Conjecture holds provided that the quotient has class at most log2(p − 2) + 1, or p ≥ 5 and G is equal to its own Baumann subgroup.

Keywords

p-groupquadratic offenderp-local finite group

MSC classification

Secondary: 20D15: Nilpotent groups, $p$-groups 20C20: Modular representations and characters
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 1 , February 2013 , pp. 211 - 222
Copyright
Copyright © Edinburgh Mathematical Society 2012

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