Finite groups having centralizer commutator product property

Abstract

Let $\alpha$ be an automorphism of a finite group $G$ and assume that $G = \big \{\,[g , \alpha ] : g \in G\,\big \}\cdot C_G (\alpha )$. We prove that the order of the subgroup $[G, \alpha ]$ is bounded above by $n^{\log _2 (n+1)}$ where $n$ is the index of $C_G (\alpha )$ in $G$.