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

Suppose that a finite group G admits a Frobenius group FH of automorphisms of coprime order with kernel F and complement H. For the case where G is a finite p-group such that G = [G, F], it is proved that the order of G is bounded above in terms of the order of H and the order of the fixed-point subgroup C (G) (H) of the complement, while the rank of G is bounded above in terms of |H| and the rank of C (G) (H). Earlier, such results were known under the stronger assumption that the kernel F acts on G fixed-point-freely. As a corollary, for the case where G is an arbitrary finite group with a Frobenius group FH of automorphisms of coprime order with kernel F and complement H, estimates are obtained which are of the form |G| a parts per thousand currency sign |C (G) (F)| center dot f(|H|, |C (G) (H)|) for the order, and of the form 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.