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Frobenius-like groups as groups of automorphisms

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. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2014) 38: 965 – 976 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1403-62 Frobenius-like groups as groups of automorphisms 1 2 ˙ ¨ ˘ G¨ ulin ERCAN1,∗, Ismail S ¸ uayip GULO GLU , Evgeny KHUKHRO3 Department of Mathematics, Middle East Technical University, Ankara, Turkey 2 ˙ Department of Mathematics, Do˘ gu¸s University, Istanbul, Turkey 3 Evgeny Khukhro, Sobolev Inst. Math., Novosibirsk, Russia Received: 21.03.2014 • Accepted: 22.05.2014 • Published Online: 24.10.2014 • Printed: 21.11.2014 Abstract: A finite group F H is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that F H/[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 F H aiming at restrictions on G in terms of CG (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. Key words: Frobenius group, Frobenius-like group, fixed points, Fitting height, nilpotency class, derived length, rank, order 1. Introduction Every nonnilpotent finite group contains nilpotent subgroups that are normalized but not centralized by elements of coprime order. Therefore, there are sections of the form 1 ̸= [N, g]⟨g⟩, where N is a nilpotent p′ -subgroup and g has prime order p . Such a section is a special case of a so-called Frobenius-like group, the formal definition of which