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

Abstract

A finite group [Formula: see text] is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup [Formula: see text] with a nontrivial complement [Formula: see text] such that [Formula: see text] for all nonidentity elements [Formula: see text]. Suppose that a finite group [Formula: see text] admits a Frobenius-like group of automorphisms [Formula: see text] of coprime order with [Formula: see text] In case where [Formula: see text] we prove that the groups [Formula: see text] and [Formula: see text] have the same nilpotent length under certain additional assumptions.

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].…”