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

<p>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| ? |C G (F)| · f(|H|, |C G (H)|) for the order, and of the form 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. © 2013 Springer Science+Business Media New York.</p>