It is shown that the number of ordered k-sets of a group G whose nth power contains exactly i elements is always a multiple of IGI. An elementary proof of the fact that the number of ordered pairs ( x , y ) such that x2 = y2 is equal to kr lGI is also given.
Vanderkam, Jeffery, "Divisibility by |G| for Powers of Ordered k-sets" (1992). Mathematical Sciences Technical Reports (MSTR). 136.