Rank and Order of a Finite Group Admitting a Frobenius-Like Group of Automorphisms

A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that FH/[F,F] is a Frobenius group with Frobenius kernel F/[F,F]. Suppose that a finite group G admits a Frobenius-like group of automorphisms FH of coprime order with certain additional restrictions (which are satisfied, in particular, if either |FH| is odd or |H| = 2). In the case where G is a finite p-group such that G = [G, F] it is proved that the rank of G is bounded above in terms of |H| and the rank of the fixed-point subgroup C (G) (H), and that |G| is bounded above in terms of |H| and |C (G) (H)|. As a corollary, in the case where G is an arbitrary finite group estimates are obtained of the form |G| a parts per thousand currency sign|C (G) (F)| center dot f(|H|, |C (G) (H)|) for the order, and r(G) a parts per thousand currency sign r(C (G) (F)) + g(|H|, r(C (G) (H))) for the rank, where f and g are some functions of two variables