On the influence of the fixed points of an automorphism to the structure of a group

Let α be a coprime automorphism of a group G of prime order and let P be an α-invariant Sylow p-subgroup of G. Assume that p∉π(CG(α)). Firstly, we prove that G is p-nilpotent if and only if CNG(P)(α) centralizes P. In the case that G is Sz(2r) and PSL(2,2r)-free where r=|α|, we show that G is p-closed if and only if CG(α) normalizes P. As a consequence of these two results, we obtain that G≅P×H for a group H if and only if CG(α) centralizes P. We also prove a generalization of the Frobenius p-nilpotency theorem for groups admitting a group of automorphisms of coprime order.