Frobenius action on Carter subgroups

Abstract

Let G be a finite solvable group and H be a subgroup of Aut(G). Suppose that there exists an H-invariant Carter subgroup F of G such that the semidirect product FH is a Frobenius group with kernel F and complement H. We prove that the terms of the Fitting series of C G (H) are obtained as the intersection of C G (H) with the corresponding terms of the Fitting series of G, and the Fitting height of G may exceed the Fitting height of C G (H) by at most one. As a corollary it is shown that for any set of primes π, the terms of the π-series of C G (H) are obtained as the intersection of C G (H) with the corresponding terms of the π-series of G, and the π-length of G may exceed the π-length of C G (H) by at most one. These theorems generalize the main results in [E. I. Khukhro, Fitting height of a finite group with a Frobenius group of automorphisms, J. Algebra 366 (2012) 1-11] obtained by Khukhro.