Finite Groups Which Are the Union of Autocentralizers of Some Automorphisms

Abstract

Let $\alpha$ be an automorphism of a given group $G$ ; then, $C_G\left(\alpha \right)=\left\{g\in G|\alpha \left(g\right)=g\right\}$ is called the autocentralizer of $\alpha$ in $G$ . In this work, we study finite groups $G$ , which can be written as the union of autocentralizers of some automorphisms of $G$ . In particular, if a group $G$ is 5- $A$ -centralizers, then we determine the absolute central quotient $G/L\left(G\right)$ of $G$ , where $A=\text{Aut}\left(G\right)$ . Finally, it is shown that autocommutative property of a group is equivalent to the one, in which every $A$ -centralizer of its nontrivial elements is abelian.