Frobenius-like groups as groups of automorphisms
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KünyeERCAN, G., GÜLOĞLU, İ.Ş., KHUKHRO, E. (2014). Frobenius-Like groups as groups of automorphisms. Turkish Journal of Mathematics, 38, pp. 965-976. http://dx.doi.org/10.3906/mat-1403-62.
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.