Derived length of a Frobenius-like kernel

A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F called kernel which has a nontrivial complement H such that F H/F, F is a Frobenius group with Frobenius kernel F/F, F. Suppose that a Frobenius-like group FH acts faithfully by linear transformations on a vector space V over a field of characteristic that does not divide |F H|. It is proved that the derived length of the kernel F is bounded solely in terms of the dimension m = dimC V(H) of the fixed-point subspace of H by g(m) = 3 + log2(m + 1). It follows that if a Frobenius-like group FH acts faithfully by coprime automorphisms on a finite group G, then the derived length of the kernel F is at most g(r), where r is the sectional rank of C G(H). As an application, for a finite solvable group G admitting an automorphism φ of prime order coprime to |G|, a bound for the p-length of G is obtained in terms of the rank of a Hall p '-subgroup of C G(φ). Earlier results of this kind were known only in the special case when the complement of the acting Frobenius-like group was assumed to have prime order and its fixed-point subspace (or subgroup) was assumed to be one-dimensional (or have all Sylow subgroups cyclic). © 2014