Nonabelian Groups with Perfect Order Subsets

Let G be a finite group and let x ∈ G. Define the order subset of G determined by x to be the set of all elements in G that have the same order as x. A group G is said to have perfect order subsets if the number of elements in each order subset of G is a divisor of |G|. In this article we prove a theorem for a class of nonabelian groups, which is analogous to Theorem 4 in [2]. We then prove that there are infinitely many nonabelian groups with perfect order subsets. In addition, all values of q are determined such that the special linear group, SL(2,q), has perfect order subsets. Next, we give a discussion of some necessary conditions for general nonabelian groups to have perfect order subsets. We conclude by stating some conjectures.