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

Given a non-abelian nite group g, let π(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 π(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 iz(g)l = lz(h)l, 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.