Frobenius-like groups as groups of automorphisms


Ercan G. , Güloğlu İ. Ş. , Khukhro E.

TURKISH JOURNAL OF MATHEMATICS, vol.38, pp.965-976, 2014 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 38
  • Publication Date: 2014
  • Doi Number: 10.3906/mat-1403-62
  • Title of Journal : TURKISH JOURNAL OF MATHEMATICS
  • Page Numbers: pp.965-976

Abstract

A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that FH/[F,F] is a Frobenius group with Frobenius kernel F/[F, F]. Such subgroups and sections are abundant in any nonnilpotent finite group. We discuss several recent results about the properties of a finite group G admitting a Frobenius-like group of automorphisms FH aiming at restrictions on G in terms of C-G(H) and focusing mainly on bounds for the Fitting height and related parameters. Earlier such results were obtained for Frobenius groups of automorphisms; new theorems for Frobenius-like groups are based on new representation-theoretic results. Apart from a brief survey, the paper contains the new theorem on almost nilpotency of a finite group admitting a Frobenius-like group of automorphisms with fixed-point-free almost extraspecial kernel.