ON THE RELATION BETWEEN THE NON-COMMUTING GRAPH AND THE PRIME GRAPH

Abstract. Given a non-abelian finite group G, let (G) denote the set of prime divisors of the order of G and denote by Z(G) the center of G. The prime graph of G is the graph with vertex set (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 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 ? 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.