Product and Quotient Set Dominance in Finite Groups

A classic problem in additive number theory asks whether a given subset of integers has more sums or differences. This inquiry naturally extends to finite groups, where one compares the sizes of the product set $AA$ and the quotient set $A{A}^{-1}$ for a subset $A\subseteq G$. A finite group is considered product-dominant if it contains a higher proportion of subsets where $\mathrm{\mid }AA\mathrm{\mid }>\mathrm{\mid }A{A}^{-1}\mathrm{\mid }$, and quotient-dominant if the reverse is true. We establish statistical criteria to determine the dominance behavior of large finite groups by analyzing the moments of the random variable $d(A):=\mathrm{\mid }AA\mathrm{\mid }-\mathrm{\mid }A{A}^{-1}\mathrm{\mid }$. By translating product and quotient relationships into graph-theoretic constructions, we compute these moments through counting independent sets. Our main result demonstrates that for a growing sequence of finite groups $G_k$ with center size $Z_k$ and involution count $T_k$, the condition $Z_k=o(T_k)$ implies the groups are eventually product-dominant. Conversely, if $T_k\cdot {\sqrt{3}}^{R_k\mathrm{/}{Z}_k}=o(Z_k)$ (where $R_k$ denotes the number of elements whose square is central), the groups are eventually quotient-dominant. Using these asymptotic conditions, we resolve the Miller–Vissuet conjecture by proving that all dihedral groups $D_n$ for $n\ge 3$ are product-dominant. Furthermore, we classify the dominance behavior of several major families of groups, proving that symmetric groups and non-abelian finite simple groups are product-dominant, while groups of odd order and certain abelian groups are quotient-dominant. Finally, for groups that fall outside our asymptotic conditions, we derive an explicit formula for the first moment utilizing the irreducible characters and conjugacy classes of the group.