Bounds on Squares of Two-Sets

Let \(G\) be a finite group and let \(p_i(G)\) denote the proportion of \((x,y) \in G^2\) for which the set \(\{x^2, xy, yx, y^2\}\) has cardinality \(i\). We show that either \(0 < p_1(G) + p_2(G) \leq \frac{1}{2}\) or \(p_1(G) + p_2(G) = 1\), and that either \(p_4(G) = 0\) or \(\frac{5}{32} \leq p_4(G) < 1\). Each of the preceding inequalities are the best possible.