Classification of Finite Groups With Equal Left and Right Quotient Sets

TL;DR

The paper resolves which finite groups $G$ satisfy $|AA^{-1}|=|A^{-1}A|$ for every subset $A\subseteq G$ (balanced groups), a problem linked to MSTD phenomena. It proves that balanced groups fall into three structural families: (i) direct products with a Hamiltonian $2$-group, (ii) sign semidirect products $C_{2^n}\ltimes Q$ with $Q\in\{C_3,C_5\}$, and (iii) non-abelian $2$-groups; from these, the exact complete list of balanced groups is obtained: finite abelian groups, $Q_8\times(C_2)^n$ ($n\ge0$), $D_6$, $D_8$, $D_{10}$, $Q_{12}$, $Q_{16}$, $C_4\ltimes C_4$, and $Q_{20}$. A notable finding is the infinite family $Q_8\times(C_2)^n$ of non-abelian balanced groups, established via anti-commutation arguments, with further filtration provided by weakly balanced concepts. The classification combines structural group-theoretic arguments with computational verification (SageMath) and subquotient stability, yielding a complete map of balanced finite groups and guiding future questions on weakly balanced and infinite cases.

Abstract

In this paper, we classify all finite groups $G$ which have the following property: for all subsets $A \subseteq G$, we have $|AA^{-1}| = |A^{-1}A|$. This question is motivated by the problem in additive combinatorics of More Sums Than Difference sets and answers several questions posed in arXiv:2509.00611 [math.NT].

Theorems & Definitions (45)

• Theorem 1.1
• Remark 1.2
• Definition 2.1
• Definition 2.2
• Theorem 2.3: baer
• Proposition 2.4
• proof
• Proposition 3.1
• proof
• Definition 3.2