Counting squares of n-subsets in finite groups

Let G be a finite group of order n ≥ 2, (cursive Greek chi1,...,cursive Greek chin) an n-tuple of elements of G and A = (aij) a square matrix of order n such that aij = cursive Greek chiicursive Greek chij. We investigate how many different types of such matrices could exist for n = 2,3 and we deal with some of their properties. We show that for every group G the number of the ordered n-tuples corresponding to the same matrix is a multiple of |G|.