Finite Groups Which Are the Union of Autocentralizers of Some Automorphisms

Abstract

Let α be an automorphism of a given group G; then, CGα=g∈G|αg=g is called the autocentralizer of α 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/LG of G, where A=AutG. 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.