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 FH/[F,F]FH/[F,F] is a Frobenius group with Frobenius kernel F/[F,F]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 |FH||FH|. It is proved that the derived length of the kernel F is bounded solely in terms of the dimension m=dimCV(H) of the fixed-point subspace of H by g(m)=3+[log2(m+1)]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)g(r), where r is the sectional rank of CG(H)CG(H). As an application, for a finite solvable group G admitting an automorphism φ of prime order coprime to |G||G|, a bound for the p -length of G is obtained in terms of the rank of a Hall p′p′-subgroup of CG(φ)CG(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)