On the structure of finite groups associated to regular non-centralizer graphs

Abstract

The non-centralizer graph of a finite group $ G $ is the simple graph $ \Upsilon_G $ whose vertices are the elements of $ G $ with two vertices are adjacent if their centralizers are distinct. The induced non-centralizer graph of $ G $ is the induced subgraph of $ \Upsilon_G $ on $ G\setminus Z(G) $. A finite group is called regular (resp. induced regular) if its non-centralizer graph (resp. induced non-centralizer graph) is regular. In this paper we study the structure of regular groups and induced regular groups. We prove that if a group $ G $ is regular, then $ G/Z(G) $ as an elementary $ 2 $-group. Using the concept of maximal centralizers, we succeeded in proving that if $ G $ is induced regular, then $ G/Z(G) $ is a $ p $-group. We also show that a group $ G $ is induced regular if and only if it is the direct product of an induced regular $ p $-group and an abelian group.

Keywords

• centralizers
• finite groups
• graph
• regular