Rank and order of a finite group admitting a Frobenius-like group of automorphisms

Abstract

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| ≤|C G (F)| · f(|H|, |C G (H)|) for the order, and r(G) ≤ r(C G (F)) + g(|H|, r(C G (H))) for the rank, where f and g are some functions of two variables.