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Mathematische Zeitschrift

, Volume 284, Issue 3–4, pp 1035–1052 | Cite as

Large \({\varvec{p}}^{\prime }\)-orbits for \({\varvec{p}}\)-nilpotent linear groups

  • David GluckEmail author
Article
  • 172 Downloads

Abstract

Let G be a p-nilpotent linear group on a finite vector space V of characteristic p. Suppose that |G||V| is odd. Let P be a Sylow p-subgroup of G. We show that there exist vectors \(v_1\) and \(v_2\) in V such that \(C_G(v_1) \cap C_G(v_2)=P\). A striking conjecture of Malle and Navarro offers a simple global criterion for the nilpotence (in the sense of Broué and Puig) of a p-block of a finite group. Our result implies that this conjecture holds for groups of odd order.

Keywords

p-Nilpotent linear groups Orbit sizes Nilpotent blocks 

Mathematics Subject Classification

20C20 20C15 20H30 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsWayne State UniversityDetroitUSA

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