A nonabelian twist on differences of bijections

Mohsen Aliabadi

arxiv: 2605.16478 · v1 · pith:JFGSCDK5new · submitted 2026-05-15 · 🧮 math.GR · math.CO

A nonabelian twist on differences of bijections

Mohsen Aliabadi This is my paper

Pith reviewed 2026-05-19 21:34 UTC · model grok-4.3

classification 🧮 math.GR math.CO

keywords Hall's theoremnonabelian groupsquotientsbijectionstilingS3abelianizationproduct-one words

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The pith

Quotient-realizability in nonabelian groups requires a cycle-tiling decomposition of partial products beyond the abelianization condition.

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

The paper extends Hall's theorem from abelian groups, where differences of two enumerations of G are possible precisely when their sum is zero, to the nonabelian setting by replacing differences with quotients. It proves that the natural necessary condition obtained by passing to the abelianization is insufficient, even when the multiset admits an ordering whose product is the identity. The central result equates the existence of bijections b and c such that A equals the multiset of all b(i)c(i)^{-1} to the existence of a decomposition of A into product-one words whose successive partial-product sets form a tiling of G by right translates. This reveals structural obstructions that are invisible to abelian invariants and are realized explicitly in S3 and infinitely many other groups.

Core claim

Quotient-realizability is equivalent to a decomposition of A into product-one words whose partial-product sets tile G by right translates.

What carries the argument

The cycle-tiling criterion that converts quotient-realizability into an exact tiling condition on the partial-product sets of product-one cycles.

If this is right

The abelianization product condition is necessary but not sufficient for a multiset to arise as quotients from two enumerations of G.
A concrete counterexample exists in the symmetric group S3.
The same obstruction extends to infinitely many finite nonabelian groups.

Where Pith is reading between the lines

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

The tiling viewpoint may classify realizable multisets in other small nonabelian groups such as dihedral or quaternion groups.
Analogous partial-product tiling conditions could appear in problems involving noncommutative representations or ordered factorizations.

Load-bearing premise

That translating the problem into permutation cycles and an exact tiling condition on partial-product sets captures every obstruction without requiring additional group-specific invariants.

What would settle it

A multiset A of size |G| in S3 that satisfies the abelianization product condition, admits a product-one ordering, and whose partial-product sets tile G under some cycle decomposition, yet cannot be realized as quotients b(i)c(i)^{-1} for any bijections b and c.

Figures

Figures reproduced from arXiv: 2605.16478 by Mohsen Aliabadi.

**Figure 2.** Figure 2: The two right-coset partitions used in the obstruction in [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

read the original abstract

Hall's theorem on differences of bijections characterizes the multisets $$ \{a_1,\ldots,a_{|G|}\} $$ in a finite abelian group $G$ that can be written in the form $$ a_i=b_i-c_i, $$ where both $b_1,\ldots,b_{|G|}$ and $c_1,\ldots,c_{|G|}$ are enumerations of $G$. The necessary and sufficient condition is the zero-sum condition $$ a_1+\cdots+a_{|G|}=0. $$ This paper studies the corresponding problem for finite nonabelian groups, with differences replaced by quotients. Thus we ask when a multiset $A$ of cardinality $|G|$ can be represented as $$ A=\{b(i)c(i)^{-1}:1\le i\le |G|\}, $$ where $b$ and $c$ are bijections onto $G$. Passing to the abelianization gives a necessary condition, namely that the product of the images of the elements of $A$ is trivial in $ G_{\rm ab}. $ We show that this condition is not sufficient in general, even when the elements of $A$ admit an ordering whose product is the identity in $G$. The main structural result is a cycle-tiling criterion: quotient-realizability is equivalent to a decomposition of $A$ into product-one words whose partial-product sets tile $G$ by right translates. The use of permutation cycles is standard, but the criterion translates quotient-realizability into an exact tiling condition. We then use this criterion to construct a counterexample in $ S_3, $ and we extend the same obstruction to infinitely many finite nonabelian groups.

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.

Desk Editor's Note private letter to a colleague

The cycle-tiling criterion gives a workable structural test for quotient-realizability beyond the abelian case, backed by a concrete S3 counterexample that the abelianization condition misses.

read the letter

The main thing to know is that this paper replaces differences with quotients and finds a cycle-tiling condition that characterizes exactly when a multiset A of size |G| can be written as b(i)c(i)^{-1} for bijections b and c on a finite nonabelian group G. Necessity comes from the cycles of the induced permutation, and they show the abelianization product condition is not enough even when an ordering multiplies to the identity. The S3 counterexample is explicit and extends to an infinite family of groups. That part is new and does not collapse to the classical Hall theorem for abelian groups. The tiling view using partial-product sets and right translates is a clean translation of the problem and gives a concrete test that combinatorial group theorists can actually apply. The necessity direction follows standard permutation-cycle arguments and looks solid. Sufficiency requires building global bijections from the given decomposition and tiling; the stress-test note worries that non-commuting elements could misalign the right translates when you try to define b and c across all cycles at once. The paper claims the equivalence holds, so presumably the construction works, but that direction is where any gap would sit and would benefit from an explicit check in the S3 case. This is for readers already working on generalizations of Hall-type results or tiling conditions in groups. It is narrow in scope but the criterion is usable and the counterexample is clear, so it deserves a serious referee to verify the sufficiency construction and the details of the infinite-family extension.

Referee Report

2 major / 2 minor

Summary. The paper studies the nonabelian version of Hall's theorem on differences of bijections, replacing additive differences with quotients: for a finite group G and multiset A of size |G|, when does there exist bijections b, c: G → G such that A = {b(i) c(i)^{-1} : i ∈ G}. It shows that the necessary condition obtained by passing to the abelianization (product of images of A is trivial in G_ab) is insufficient in general, even when A admits an ordering with product equal to the identity in G. The central result is a cycle-tiling criterion equating quotient-realizability to a decomposition of A into product-one words whose partial-product sets tile G by right translates; this is applied to produce an explicit counterexample in S_3 and to exhibit the same obstruction in infinitely many finite nonabelian groups.

Significance. If the cycle-tiling equivalence is valid, the paper supplies a useful combinatorial characterization that isolates obstructions beyond the abelianization product and demonstrates that nonabelian phenomena genuinely appear. The explicit S_3 counterexample and its extension to other groups are concrete contributions that clarify the gap between the abelian and nonabelian settings.

major comments (2)

[main structural result / cycle-tiling theorem] Main structural result (cycle-tiling criterion): the sufficiency direction asserts that any decomposition of A into product-one words whose partial-product sets tile G by right translates yields realizing bijections b and c. In the nonabelian case the right translates S g = {s g} need not remain disjoint or cover G after the successive left or right multiplications by elements of A are performed across all cycles, because non-commuting elements can misalign the translates. The manuscript must supply an explicit global construction of b and c (or a proof that the tiling condition forces the required disjointness) rather than relying on the abelian intuition that right translates automatically produce a permutation.
[S_3 counterexample] § on the S_3 counterexample: the verification that the chosen multiset satisfies the tiling condition yet fails to be quotient-realizable must be checked against the precise definition of the partial-product sets; any ambiguity in how the right translates are indexed by the cycles could affect whether the obstruction is genuinely new or reducible to the abelianization condition already known to be insufficient.

minor comments (2)

[preliminaries / notation] Notation for partial-product sets and right translates should be introduced with a short diagram or explicit indexing to avoid confusion when the group is nonabelian.
[introduction] The statement that the abelianization condition is 'not sufficient even when the elements admit an ordering whose product is the identity' would benefit from a brief remark on why the ordering condition is mentioned separately from the abelianization product.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address each major comment below and will incorporate clarifications and additional details in a revised version.

read point-by-point responses

Referee: [main structural result / cycle-tiling theorem] Main structural result (cycle-tiling criterion): the sufficiency direction asserts that any decomposition of A into product-one words whose partial-product sets tile G by right translates yields realizing bijections b and c. In the nonabelian case the right translates S g = {s g} need not remain disjoint or cover G after the successive left or right multiplications by elements of A are performed across all cycles, because non-commuting elements can misalign the translates. The manuscript must supply an explicit global construction of b and c (or a proof that the tiling condition forces the required disjointness) rather than relying on the abelian intuition that right translates automatically produce a permutation.

Authors: We appreciate this observation on the nonabelian setting. The cycle-tiling condition is formulated so that the partial-product sets for each cycle are chosen to be disjoint right translates that together cover G; the product-one closure of each cycle then ensures that the assignments of b and c remain consistent and bijective without overlap, even when elements fail to commute. To eliminate any reliance on abelian intuition, we will add an explicit inductive construction of the bijections b and c in the revised manuscript: label the elements of G by the tiled partial-product sets, then define b and c along each cycle by shifting according to the word in A while using the fixed right translates to guarantee uniqueness. revision: yes
Referee: [S_3 counterexample] § on the S_3 counterexample: the verification that the chosen multiset satisfies the tiling condition yet fails to be quotient-realizable must be checked against the precise definition of the partial-product sets; any ambiguity in how the right translates are indexed by the cycles could affect whether the obstruction is genuinely new or reducible to the abelianization condition already known to be insufficient.

Authors: We agree that a fully explicit verification is needed. In the revision we will list the cycles, the associated product-one words from A, the precise partial-product sets, and the right translates for each, confirming both that they form a partition of S_3 and that no pair of bijections b, c realizes the multiset. This will also make clear that the obstruction is independent of the abelianization product condition. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the cycle-tiling equivalence derivation

full rationale

The paper establishes an equivalence between quotient-realizability of a multiset A and a decomposition into product-one words whose partial-product sets tile G by right translates. Necessity is derived directly from the cycles of the permutation induced by any realizing pair of bijections b and c, with partial products and right translates following immediately from the definition of the quotients b(i)c(i)^{-1}. Sufficiency is addressed via an explicit converse construction that defines global bijections from the given decomposition and tiling. The abelianization product condition is presented only as a necessary (but not sufficient) consequence, with a separate counterexample constructed in S3. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the derivation relies on standard group-theoretic techniques applied to the problem's own definitions without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard finite group theory and abelianization without introducing new free parameters or invented entities; the tiling condition is derived rather than postulated.

axioms (1)

standard math Standard properties of finite groups, quotients, and abelianization maps
Invoked when passing to G_ab to obtain the necessary product condition.

pith-pipeline@v0.9.0 · 5835 in / 1382 out tokens · 44663 ms · 2026-05-19T21:34:26.851803+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

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Gao and A

W. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: A survey,Expo. Math.24 (2006), no. 4, 337–369. 17

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Geroldinger and D

A. Geroldinger and D. J. Grynkiewicz, The large Davenport constant. I. Groups with a cyclic, index two subgroup,J. Pure Appl. Algebra217(2013), no. 5, 863–885

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Hall Jr., A combinatorial problem on abelian groups,Proc

M. Hall Jr., A combinatorial problem on abelian groups,Proc. Amer. Math. Soc.3(1952), 584–587

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M. Hall Jr. and L. J. Paige, Complete mappings of finite groups,Pacific J. Math.5(1955), 541–549

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Salzborn and G

F. Salzborn and G. Szekeres, A problem in combinatorial group theory,Ars Combin.7(1979), 3–5

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D. H. Ullman and D. J. Velleman, Differences of bijections,Amer. Math. Monthly126(2019), no. 3, 199–216

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I. M. Wanless, Transversals in Latin squares: A survey, inSurveys in Combinatorics 2011, London Math. Soc. Lecture Note Ser., vol. 392, Cambridge University Press, Cambridge, 2011, pp. 403–437. 18

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[1] [1]

Alspach, J.-C

B. Alspach, J.-C. Bermond, and D. Sotteau, Decomposition into cycles. I. Hamilton decompositions, in Cycles and Rays, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 301, Kluwer Academic Publishers, Dordrecht, 1990, pp. 9–18

work page 1990

[2] [2]

D´ enes and A

J. D´ enes and A. D. Keedwell,Latin Squares and Their Applications, Academic Press, New York, 1974

work page 1974

[3] [3]

A. B. Evans, Applications of complete mappings and orthomorphisms of finite groups,Quasigroups Related Systems23(2015), no. 1, 5–30

work page 2015

[4] [4]

Fuchs, Ein kombinatorisches Problem bez¨ uglich abelscher Gruppen,Math

L. Fuchs, Ein kombinatorisches Problem bez¨ uglich abelscher Gruppen,Math. Nachr.18(1958), 292–297

work page 1958

[5] [5]

Gao and A

W. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: A survey,Expo. Math.24 (2006), no. 4, 337–369. 17

work page 2006

[6] [6]

Geroldinger and D

A. Geroldinger and D. J. Grynkiewicz, The large Davenport constant. I. Groups with a cyclic, index two subgroup,J. Pure Appl. Algebra217(2013), no. 5, 863–885

work page 2013

[7] [7]

Hall Jr., A combinatorial problem on abelian groups,Proc

M. Hall Jr., A combinatorial problem on abelian groups,Proc. Amer. Math. Soc.3(1952), 584–587

work page 1952

[8] [8]

M. Hall Jr. and L. J. Paige, Complete mappings of finite groups,Pacific J. Math.5(1955), 541–549

work page 1955

[9] [9]

M. A. Ollis, Sequenceable groups and related topics,Electron. J. Combin.19(2012), Dynamic Survey DS10

work page 2012

[10] [10]

Salzborn and G

F. Salzborn and G. Szekeres, A problem in combinatorial group theory,Ars Combin.7(1979), 3–5

work page 1979

[11] [11]

D. H. Ullman and D. J. Velleman, Differences of bijections,Amer. Math. Monthly126(2019), no. 3, 199–216

work page 2019

[12] [12]

I. M. Wanless, Transversals in Latin squares: A survey, inSurveys in Combinatorics 2011, London Math. Soc. Lecture Note Ser., vol. 392, Cambridge University Press, Cambridge, 2011, pp. 403–437. 18

work page 2011