Some results on characterization of finite group by non commuting graph

Abstract

The non commuting graph of a non-abelian finite group $G$ is defined as follows: its vertex set is $G-Z(G)$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if the commutator of $x$ and $y$ is not the identity. In this paper we prove some new results about this graph. In particular we will give a new proof of theorem 3.24 of [2]. We also prove that if $G_1$, $G_2$, ..., $G_n$ are finite groups such that $Z(G_i)=1$ for $i=1,2,...,n$ and they are characterizable by non commuting graph, then $G_1times ...times G_n$ is characterizable by non commuting graph.

Keywords

• non commuting graph
• nilpotent groups
• Finite groups