Fitting Height of a Finite Group with a Metabelian Group of Automorphisms

Abstract

Let M = FH be a finite group that is a product of a normal abelian subgroup F and an abelian subgroup H. Assume that all elements in M\F have prime order p, and F has at most one subgroup of order p. Examples of such groups are dihedral groups for p = 2 and the semidirect product of a cyclic group F by a

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.…”

“…In [2] we can find an example which shows that for odd prime the equality F i (C G (α)) = C G (α) ∩ F i (G) need not hold. It should be also mentioned that by Turull [13] we already have that h(G) ≤ h(C G (α)) + 2.…”

“…In the present work we show that the techniques developed in [2] and [8] can be used in the study of actions by p-groups of maximal class with a fixed-point-free maximal subgroup. We denote by h(G) the Fitting height of a group G and by F i (G) the i-th term of its Fitting series.…”