COUNTEREXAMPLES FOR SOME CONJECTURES OF FINITE GROUPS

For a finite group G, let e (G) be the set of orders of its elements.The order classes of G, denoted as OC(G), are collections of pairs [k, m],where k e (G) and m represents the number of elements in G with order k.Two finite groups G and H are said to be of the same order type if OC(G) = OC(H).It is unequivocal that two simple groups G 1 and G 2 with the same order classes are isomorphic.This paper presents counterexamples demonstrating that this property does not necessarily hold for non-simple groups.Furthermore, we will examine several properties of non-simple groups that derived from their group order and element orders, and address certain open problems related to these properties.