On Groups with Perfect Order Subsets

A finite group G is said to have Perfect Order Subsets if for every d, the number of elements of G of order d (if there are any) divides |G|. This notion was introduced in the paper [1] by C. Finch and L. Jones. In the case of finite Abelian groups, the authors reduced the problem of which groups have this property to the case of groups of the form G = ∏k i=1(Z/piZ) ai , where pi are primes and ai > 1. For these groups, it follows from results in [1] thatG has Perfect Order Subsets if and only if f(n)|n. Only 11 examples of such n are known, given below, and only one of these is divisible by the square of an odd prime.