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

Abstract: Let $\alpha$ be a coprime automorphism of a group $G$ of prime order and let $P$ be an $\alpha$-invariant Sylow $p$-subgroup of $G$. Assume that $p\notin \pi(C_G(\alpha))$. Firstly, we prove that $G$ is $p$-nilpotent if and only if $C_{N_G(P)}(\alpha)$ centralizes $P$. In the case that $G$ is $Sz(2^r)$ and $PSL(2,2^r)$-free where $r=|\alpha|$, we show that $G$ is $p$-closed if and only if $C_G(\alpha)$ normalizes $P$. As a consequences of these two results, we obtain that $G\cong P\times H$ for a group $H$ if and only if $C_G(\alpha)$ centralizes $P$. We also prove a generalization of the Frobenius $p$-nilpotency theorem for groups admitting a group of automorphisms of coprime order.