Frobenius-like groups as groups of automorphisms

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 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 kerne

On p-groups of maximal class as groups of automorphisms

“…For example, there are recent results obtained by Ercan, Güloğlu and Khukhro where the condition of BA being Frobenius was replaced by BA/ [B, B] being Frobenius, see for instance [3][4][5]. Earlier in [11], Shumyatsky also extended for dihedral groups some results proved for Frobenius groups.…”

“…Although the original definition of uniform elements are only for p-groups of order greater than p 4 , in what follows we say that an element in a p-group is uniform whenever its centralizer has order p 2 . Moreover, here we use the fact that we always find uniform elements outside any maximal subgroup.…”

Action of a Frobenius-like group with kernel having central derived subgroup

“…In case C W (F ) = 0 we apply Corollary 3.4 in [4] to the action of the Frobenius group (F 1 /F )H on C W (F ) and see that C W (F ) H is free. Since C W (H) = 0 by (8) we get C W (F ) = 0.…”

“…Extraspecial groups F admitting a group of automorphisms H which acts semiregularly on F/F and centralizes F provide examples F H of Frobenius-like groups which contain F in the center. In [2] and [4] we investigated the structure of solvable groups G which admit a Frobenius-like group of the above type for which F is of prime order. It is observed in the present paper that whenever the action of F on the top Fitting factor of G is Frobenius, that is, (G/F n−1 (G))F is a Frobenius group; the condition that F is of prime order can be replaced by the weaker condition that F has a maximal subgroup of prime power order.…”

Frobenius action on Carter subgroups

“…So far we have obtained several extensions of Khukhro's result replacing F H by a Frobenius-like group with kernel F under some mild additional conditions (see [3], [4], [5], [6], [7], [8], [9]). Unfortunately, this time we must be satisfied with the present form due to the example given in [4].…”