Bounds on Squares of Two-Sets

Abstract

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 $0<p_1(G)+p_2(G)\le \frac{1}{2}$ or $p_1(G)+p_2(G)=1$, and that either $p_4(G)=0$ or $\frac{5}{32}\le p_4(G)<1$. Each of the preceding inequalities are the best possible.