Frobenius action on Carter subgroups.

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 F H 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. Algebra366 (2012) 1–11] obtained by Khukhro. [ABSTRACT FROM AUTHOR]