%0 Journal Article
%T On the relation between the non-commuting graph and the prime graph
%J International Journal of Group Theory
%I University of Isfahan
%Z 2251-7650
%A Ahanjideh, Neda
%A Iranmanesh, A.
%D 2012
%\ 03/01/2012
%V 1
%N 1
%P 25-28
%! On the relation between the non-commuting graph and the prime graph
%K Non-commuting graph
%K Prime graph, Maximal abelian subgroups, Maximal independent
set of the graph
%R 10.22108/ijgt.2012.469
%X $\pi(G)$ denote the set of prime divisors of the order of $G$ and denote by $Z(G)$ the center of $G$. The\textit{ prime graph} of $G$ is the graph with vertex set $\pi(G)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $G$ contains an element of order $pq$ and the \textit{non-commuting graph} of $G$ is the graph with the vertex set $G-Z(G)$ where two non-central elements $x$ and $y$ are joined by an edge if and only if $xy \neq yx$.
Let $ G $ and $ H $ be non-abelian finite groups with isomorphic non-commuting graphs. In this article, we show that if $ | Z ( G ) | = | Z ( H ) | $, then $ G $ and $ H $ have the same prime graphs and also, the set of orders of the maximal abelian subgroups of $ G $ and $ H $ are the same.
%U https://ijgt.ui.ac.ir/article_469_92246b683ed4fb19f106761cb0e6d800.pdf