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On nilpotent subgroups containing non-trivial normal subgroups

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Veröffentlicht/Copyright: 20. November 2009
Journal of Group Theory
Aus der Zeitschrift Band 13 Heft 3

Abstract

Let G be a non-trivial finite group and let A be a nilpotent subgroup of G. We prove that if |G : A| ⩽ exp(A), the exponent of A, then A contains a non-trivial normal subgroup of G. This extends an earlier result of Isaacs, who proved this in the case where A is abelian. We also show that if the above inequality is replaced by |G : A| < Exp(G), where Exp(G) denotes the order of a cyclic subgroup of G with maximal order, then A contains a non-trivial characteristic subgroup of G. We will use these results to derive some facts about transitive permutation groups.

Received: 2009-03-18
Revised: 2009-07-25
Published Online: 2009-11-20
Published in Print: 2010-May

© de Gruyter 2010

Artikel in diesem Heft

  1. Classifying spaces of subgroups of profinite groups
  2. On spaces of Conradian group orderings
  3. The twisted conjugacy problem for endomorphisms of polycyclic groups
  4. On finite-index extensions of subgroups of free groups
  5. Detecting hypercentrality from small subgroups
  6. Warfield dualities induced by self-small mixed groups
  7. On nilpotent subgroups containing non-trivial normal subgroups
  8. Powerful actions and non-abelian tensor products of powerful p-groups
  9. On Gelfand models for finite Coxeter groups
  10. On parabolic closures in Coxeter groups
  11. On the normal index and the c-section of maximal subgroups of a finite group
https://doi.org/10.1515/jgt.2009.058

Artikel in diesem Heft

  1. Classifying spaces of subgroups of profinite groups
  2. On spaces of Conradian group orderings
  3. The twisted conjugacy problem for endomorphisms of polycyclic groups
  4. On finite-index extensions of subgroups of free groups
  5. Detecting hypercentrality from small subgroups
  6. Warfield dualities induced by self-small mixed groups
  7. On nilpotent subgroups containing non-trivial normal subgroups
  8. Powerful actions and non-abelian tensor products of powerful p-groups
  9. On Gelfand models for finite Coxeter groups
  10. On parabolic closures in Coxeter groups
  11. On the normal index and the c-section of maximal subgroups of a finite group
Heruntergeladen am 29.4.2026 von https://www.degruyterbrill.com/document/doi/10.1515/jgt.2009.058/html?lang=de
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