Counting Squares of \(n\)-Subsets in Finite Groups

Let \(G\) be a finite group of order \(n \geq 2\), \((x_1, \ldots, x_{ n})\) an \(n\)-tuple of elements of \(G\) and \(A = (a_{ij})\) a square matrix of order \(n\) such that \(a_{ij} = x_ix_j\). 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 \(|G|\).