CHARACTERIZATION OF THE SYMMETRIC GROUP BY ITS NON-COMMUTING GRAPH

Abstract. The non-commuting graph r(G) of a non-abelian 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 that if G is a finite group with r(G) = r(Sn), then G = Sn, where Sn is the symmetric group of degree n, where n is a natural number.