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 vertexset is g - z(g) and two distinct vertices x and y are joined by an edge if and only if the commutatorof 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.