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arxiv: 2509.00611 · v3 · submitted 2025-08-30 · 🧮 math.NT · math.CO

Comparing Left and Right Quotient Sets in Groups

Julian Duvivier , Xiaoyao Huang , Ava Kennon , Say-Yeon Kwon , Steven J. Miller , Arman Rysmakhanov , Pramana Saldin , Ren Watson This is my paper

Pith reviewed 2026-05-18 19:06 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords quotient setscardinality differenceinfinite dihedral groupfree groupdifference graphsnon-abelian groupsleft and right quotients
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The pith

Every integer difference is achievable in the infinite dihedral group but only even differences in the free group on two generators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how the cardinalities of the right quotient set AA^{-1} and left quotient set A^{-1}A can differ for finite subsets A in non-abelian groups. It proves that any integer difference can be realized by choosing appropriate A in the infinite dihedral group. For the free group on two generators, the difference must be even and the authors give explicit subsets that achieve each even integer. The analysis relies on constructing graphs to count distinct quotients via their connected components. This provides a concrete way to measure non-commutativity through set sizes.

Core claim

For any integer k, there is a finite subset A of the infinite dihedral group such that the cardinality of AA^{-1} minus the cardinality of A^{-1}A equals k. In the free group F_2, this difference equals k if and only if k is even, and subsets realizing each even k are constructed explicitly.

What carries the argument

The difference graphs D_A and D_{A^{-1}}, which record equalities among the elements of the right and left quotient sets; the bijection between their edges and the count of connected components determine the exact difference in cardinalities.

If this is right

  • Any prescribed integer difference between |AA^{-1}| and |A^{-1}A| can be realized in the infinite dihedral group.
  • Only even differences between |AA^{-1}| and |A^{-1}A| can be realized in the free group on two generators.
  • Explicit constructions exist in F_2 for every even difference value.
  • The smallest size of A yielding a nonzero difference depends on the existence of order-two elements in the group.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar difference graphs could be used to study quotient set asymmetries in other classes of non-abelian groups.
  • The parity restriction observed in F_2 may stem from an underlying invariant preserved under the group multiplication.
  • These findings could inform the study of growth rates or expansion properties in groups where left and right multiplications differ.

Load-bearing premise

That the bijection between the edges of the two difference graphs and the resulting count of connected components yields the precise cardinality difference without additional coincidences.

What would settle it

A calculation showing an odd difference in F_2 or the absence of a subset realizing difference 1 in D_∞ would falsify the respective claims.

read the original abstract

For a finite subset $A$ of a group $G$, we define the right quotient set and the left quotient set of $A$, respectively, as $AA^{-1} := \{a_1a_2^{-1}:a_1,a_2\in A\}$, $A^{-1}A := \{a_1^{-1}a_2:a_1,a_2\in A\}$. While the right and left quotient sets are equal if $G$ is abelian, subtleties arise when $G$ is a nonabelian group, where the cardinality difference $|AA^{-1}| - |A^{-1}A|$ may be take on arbitrarily large values. Using the results of Martin and O'Bryant on the cardinality differences of sum sets and difference sets in $\mathbb{Z}$, we prove in the infinite dihedral group, $D_\infty \cong \mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z}$, every integer difference is achievable. Further, we prove that in $F_2$, the free group on $2$ generators, an integer difference is achievable if and only if that integer is even, and we explicitly construct subsets of $F_2$ that achieve every even integer. We further determine the minimum cardinality of $A \subset G$ so that the difference between the cardinalities of the left and right quotient sets is nonzero, depending on the existence of order $2$ elements in $G$. To prove these results, we construct difference graphs $D_A$ and $D_{A^{-1}}$ which encode equality, respectively, in the right and left quotient sets. We observe a bijection from edges in $D_A$ to edges in $D_{A^{-1}}$ and count connected components in order to obtain our results on cardinality differences $|AA^{-1}| - |A^{-1}A|$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper examines the possible integer values of the difference |AA^{-1}| − |A^{-1}A| for finite subsets A of non-abelian groups G. It proves that every integer arises in the infinite dihedral group D_∞ ≅ ℤ ⋊ ℤ/2ℤ by reducing to results of Martin–O'Bryant on sumset/difference-set differences in ℤ. In the free group F_2 it shows that only even integers arise and constructs explicit A realizing every even value. It also determines the minimal |A| making the difference nonzero, depending on the presence of order-2 elements. The proofs introduce difference graphs D_A and D_{A^{-1}} whose edges encode equalities among right (resp. left) quotients, establish an edge bijection between the two graphs, and obtain the cardinality difference from the resulting counts of connected components.

Significance. If the translation from edge bijection to component-count difference is valid, the work supplies the first complete characterizations of achievable quotient-set differences in two important non-abelian groups and introduces a graph-theoretic technique that may apply more broadly. The explicit constructions and the minimal-size results are concrete contributions to combinatorial group theory.

major comments (1)
  1. [§3 and proofs of Theorems 4.1, 5.3] §3 (Difference Graphs) and the proofs of Theorems 4.1 and 5.3: An edge bijection between D_A and D_{A^{-1}} is asserted, after which the cardinality difference |AA^{-1}| − |A^{-1}A| is deduced from the difference in the numbers of connected components. Because the vertex sets of the two graphs may be distinct (A versus A^{-1} or pairs thereof), an arbitrary set bijection on edges does not automatically determine a fixed relation between the component counts. The manuscript must state the precise formula (e.g., |AA^{-1}| = |V| − c(D_A) + k) and verify that the bijection preserves incidence sufficiently to make the component difference equal the claimed quotient difference. This step is load-bearing for both main theorems.
minor comments (2)
  1. [Abstract] Abstract, line 3: “may be take on” should read “may take on”.
  2. [References] The citation to Martin–O'Bryant should appear with a full bibliographic entry in the references section.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address the major comment point by point below and will revise the paper to improve the clarity of the graph-theoretic arguments.

read point-by-point responses

  1. Referee: [§3 and proofs of Theorems 4.1, 5.3] §3 (Difference Graphs) and the proofs of Theorems 4.1 and 5.3: An edge bijection between D_A and D_{A^{-1}} is asserted, after which the cardinality difference |AA^{-1}| − |A^{-1}A| is deduced from the difference in the numbers of connected components. Because the vertex sets of the two graphs may be distinct (A versus A^{-1} or pairs thereof), an arbitrary set bijection on edges does not automatically determine a fixed relation between the component counts. The manuscript must state the precise formula (e.g., |AA^{-1}| = |V| − c(D_A) + k) and verify that the bijection preserves incidence sufficiently to make the component difference equal the claimed quotient difference. This step is load-bearing for both main theorems.

    Authors: We agree that the manuscript should provide a more detailed explanation of how the edge bijection leads to the difference in connected components. The bijection is induced by the inversion map a ↦ a^{-1}, which provides a correspondence between the vertices of D_A and D_{A^{-1}} as well as between their edges. In the revised version, we will explicitly state the formula connecting the quotient set cardinality to the number of connected components and verify that the incidence is preserved under this map, thereby justifying the deduction of the cardinality difference. This clarification will be added to Section 3 and the relevant proofs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from explicit constructions and external theorems

full rationale

The paper derives its claims on achievable integer differences |AA^{-1}| - |A^{-1}A| in D_∞ and F_2 by constructing explicit finite subsets A and applying the external Martin-O'Bryant theorems on sumset/difference-set cardinalities in Z. The difference graphs D_A and D_{A^{-1}} are introduced to encode equalities among quotients, with an observed edge bijection used to relate connected-component counts to the target cardinality difference. This constitutes a direct combinatorial argument rather than a self-definitional loop, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The central results remain independent of the paper's own inputs and rest on verifiable constructions plus cited external results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the Martin-O'Bryant cardinality theorems for integer sum and difference sets and on the correctness of the edge-bijection and component-counting argument in the difference graphs.

axioms (1)
  • domain assumption Martin-O'Bryant results on cardinality differences of sum sets and difference sets in Z
    Invoked to transfer integer results to the infinite dihedral group.

pith-pipeline@v0.9.0 · 5906 in / 1362 out tokens · 39333 ms · 2026-05-18T19:06:45.934385+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 1 internal anchor

  1. [1]

    Sum and difference sets in generalized dihedral groups

    Ruben Ascoli, Justin Cheigh, Ryan Jeong, Andrew Keisling, Astrid Lilly, Steven J Miller, Prakod Ngamlamai, Matthew Phang, et al. Sum and difference sets in generalized dihedral groups. arXiv preprint arXiv:2210.00669 , 2022

    work page arXiv 2022

  2. [2]

    Miller, and Lily Shao

    Hung Viet Chu, Noah Luntzlara, Steven J. Miller, and Lily Shao. Infinite families of partitions into MSTD subsets. Integers: Electronic Journal of Combinatorial Number Theory , 2019

    work page 2019

  3. [3]

    Generalizations of a curious family of MSTD sets hidden by interior blocks

    H \`u ng Vi \^e t Chu, Noah Luntzlara, Steven J Miller, and Lily Shao. Generalizations of a curious family of MSTD sets hidden by interior blocks. Integers: Electronic Journal of Combinatorial Number Theory , 2020

    work page 2020

  4. [4]

    Peter V. Hegarty. Some explicit constructions of sets with more sums than differences. Acta Arithmetica , 130(1):61–77, 2007

    work page 2007

  5. [5]

    Products of subsets of groups by their inverses

    Marcel Herzog, Gil Kaplan, Patrizia Longobardi, and Mercede Maj. Products of subsets of groups by their inverses. Beitr. Algebra Geom. , 55(2):311--346, 2014

    work page 2014

  6. [6]

    Constructions of generalized mstd sets in higher dimensions

    Elena Kim and Steven J Miller. Constructions of generalized mstd sets in higher dimensions. Journal of Number Theory , 235:358--381, 2022

    work page 2022

  7. [7]

    L'addition des variables al \'e atoires d \'e finies sur une circonf \'e rence

    Paul L \'e vy. L'addition des variables al \'e atoires d \'e finies sur une circonf \'e rence. Bulletin de la Soci \'e t \'e Math \'e matique de France , 67:1--41, 1939

    work page 1939

  8. [8]

    Miller, and Kevin O'Bryant

    Oleg Lazarev, Steven J. Miller, and Kevin O'Bryant. Distribution of missing sums in sumsets. Exp. Math. , 22(2):132--156, 2013

    work page 2013

  9. [9]

    C. L \"o h. Geometric Group Theory: An Introduction . Universitext. Springer International Publishing, 2017

    work page 2017

  10. [10]

    Miller and Mark Nigrini

    Steven J. Miller and Mark Nigrini. The modulo 1 central limit theorem and Benford's law for products. International Journal of Algebra , 2(3):119--130, 2008

    work page 2008

  11. [11]

    Many sets have more sums than differences

    Greg Martin and Kevin O'Bryant. Many sets have more sums than differences. arXiv preprint math/0608131 , 2006

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  12. [12]

    Classification of finite groups with equal left and right quotient sets, 2025

    Haran Mouli and Pramana Saldin. Classification of finite groups with equal left and right quotient sets, 2025

    work page 2025

  13. [13]

    Most subsets are balanced in finite groups

    Steven J Miller and Kevin Vissuet. Most subsets are balanced in finite groups. In Combinatorial and Additive Number Theory: CANT 2011 and 2012 , pages 147--157. Springer, 2014

    work page 2011

  14. [14]

    Sets with more sums than differences

    Melvyn B Nathanson. Sets with more sums than differences. Integers: Electronic Journal of Combinatorial Number Theory , 7(A05):A05, 2007

    work page 2007

  15. [15]

    254b, notes 5: Product theorems, pivot arguments, and the larsen--pink non-concentration inequality

    Terence Tao. 254b, notes 5: Product theorems, pivot arguments, and the larsen--pink non-concentration inequality

    work page

  16. [16]

    Product set estimates for non-commutative groups

    Terence Tao. Product set estimates for non-commutative groups. Combinatorica , 28(5):547--594, 2008

    work page 2008

  17. [17]

    S ageMath, the S age M athematics S oftware S ystem ( V ersion 10.6) , 2025

    The Sage Developers . S ageMath, the S age M athematics S oftware S ystem ( V ersion 10.6) , 2025. https://www.sagemath.org

    work page 2025

  18. [18]

    Counting MSTD sets in finite abelian groups

    Yufei Zhao. Counting MSTD sets in finite abelian groups. Journal of Number Theory , 130(10):2308--2322, 2010

    work page 2010

  19. [19]

    Sets characterized by missing sums and differences

    Yufei Zhao. Sets characterized by missing sums and differences. Journal of Number Theory , 131(11):2107–2134, November 2011

    work page 2011