Primary Cosets in Groups

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Abstract

A finite group G is called a generalized Frobenius group with kernel F if F is a proper nontrivial normal subgroup of G, and for every element Fx of prime order p in the quotient group G/F, the coset Fx of G consists of p-elements. We study generalized Frobenius groups with an insoluble kernel F. It is proved that F has a unique non- Abelian composition factor, and that this factor is isomorphic to L2(32l) for some natural number l. Moreover, we look at a (not necessarily finite) group generated by a coset of some subgroup consisting solely of elements of order three. It is shown that such a group contains a nilpotent normal subgroup of index three.

Original languageEnglish
Pages (from-to)216-221
Number of pages6
JournalAlgebra and Logic
Volume59
Issue number3
DOIs
Publication statusPublished - Jul 2020

Keywords

  • coset
  • generalized Frobenius group
  • insoluble group
  • projective special linear group

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