For a finite group G, we define an equivalence relation on G as follows: 8 x; y 2 G xy () jxGj¼jyGj; where x G is the conjugacy class of x in G. The set of sizes of equivalence classes with respect to this relation is called the same-size conjugate set of G and denote it by U(G). In this paper, we consider the influence of U(G) on the structure of a group G. In fact, we conjecture that all finite nonabelian simple groups G are characterizable by U(G) and we confirm this conjecture for the projective special linear groups PSL3(3) and PSL2(q), where q 2 f5; 7; 8; 9; 17g
