Product and Quotient Set Dominance in Finite Groups

Abstract

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 AA⁻¹ for a subset A ⊆ G. A finite group is considered product-dominant if it contains a higher proportion of subsets where |AA| > |AA⁻¹|, 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) := |AA| − |AA⁻¹|. 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 · √3^R_k/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 ≥ 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.

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BibTeX RIS

BibTeX

@misc{amit2026product,
  title = {{Product and Quotient Set Dominance in Finite Groups}},
  author = {Yuval Amit and Lucas Chen and Christopher Housholder and Joshua Im and Joshua Khan and Steven J. Miller and Mateo Palomares and Devayani Pradhan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12912},
  eprint = {2610.12912},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12912},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Product and Quotient Set Dominance in Finite Groups
AU  - Yuval Amit
AU  - Lucas Chen
AU  - Christopher Housholder
AU  - Joshua Im
AU  - Joshua Khan
AU  - Steven J. Miller
AU  - Mateo Palomares
AU  - Devayani Pradhan
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.12912
UR  - https://arxiv.org/abs/2610.12912
AB  - 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 AA^−1 for a subset A ⊆ G. A finite group is considered product-dominant if it contains a higher proportion of subsets where |AA| > |AA^−1|, 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) := |AA| − |AA^−1|. 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 · √3^R_k/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 ≥ 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.
ER  -

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