Frobenius action on Carter subgroups
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Let G he 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 pi, the terms of the pi-series of C-G (H) are obtained as the intersection of C-G (H) with the corresponding terms of the pi-series of G, and the pi-length of C may exceed the pi-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.