The all-orders classification of subsquares of group multiplication tables without transversals (Müyesser–Pokrovskiy Conjecture 7.1)

Müyesser and Pokrovskiy proved that for all sufficiently large $n$, a subsquare $A\times B$ of the multiplication table of a finite group $G$, with $|A|=|B|=n$, contains a transversal unless either $A$ and $B$ are cosets of a subgroup violating the Hall–Paige condition, or $A\times B$ is obtained from the table of an elementary abelian 2-group by deleting two rows and two columns whose elements sum to zero. Their probabilistic absorption methods are intrinsically asymptotic, and they conjecture the classification holds for every finite order. This problem asks for a proof or disproof of that all-$n$ conjecture; its cyclic even-order specialization is the second case of Snevily's 1999 conjecture, open ever since it was posed.

Background

A complete mapping of a finite group $G$ is a bijection $\phi:G\to G$ such that $x\mapsto x\phi(x)$ is also a bijection; equivalently, it is a transversal of the multiplication table $M(G)$, the Latin square whose rows and columns are indexed by the elements of $G$ and whose entry in row $g$ and column $h$ is $gh$. Here a Latin square is an $n\times n$ array filled with $n$ symbols so that each symbol appears exactly once in each row and each column, and a transversal is a selection of $n$ cells no two of which share a row, a column, or a symbol. Complete mappings were introduced by Hall and Paige (1955), who showed that a necessary condition for their existence is the Hall–Paige condition: the product of all elements of $G$ must be trivial in the abelianization $G^{\mathrm{ab}}=G/G'$ (where $G'$ is the commutator subgroup), which is equivalent to every Sylow 2-subgroup of $G$ (a subgroup whose order is the largest power of 2 dividing $|G|$) being trivial or non-cyclic. They conjectured sufficiency, confirmed in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups; Eberhard, Manners, and Mrazović (2022) later gave an analytic proof for large groups together with a precise asymptotic count of complete mappings.

A natural refinement, due to Snevily (1999), concerns subsquares: deleting rows and columns from $M(G)$ should only make transversals easier to find, so which subsquares fail to have one? Formally, for $A,B\subseteq G$ with $|A|=|B|=n$, the subsquare $A\times B$ is the $n\times n$ array with rows indexed by $A$, columns by $B$, and entry $ab$ in row $a$ and column $b$. Snevily conjectured that for abelian $G$ of odd order every subsquare has a transversal, and that for cyclic $G$ of even order the only obstructions are subsquares that are cosets of the multiplication table of an even-order cyclic subgroup. The odd-order claim was proved for prime-order cyclic groups by Alon (2000) via the Combinatorial Nullstellensatz (a polynomial-method tool for forcing solutions of systems over fields), extended to all odd-order cyclic groups by Dasgupta, Károlyi, Serra, and Szegedy (2001), and settled for all abelian groups of odd order by Arsovski (2011) using character theory. The even-order cyclic case remains open. In even order a second genuine obstruction exists: in an elementary abelian 2-group $(\mathbb{Z}_2)^k$, deleting two rows and two columns whose labelling elements sum to zero produces a subsquare with no transversal.

Müyesser and Pokrovskiy (preprint 2022, Inventiones mathematicae 2025) proved a far-reaching common generalization for large groups. Their main theorem, proved with probabilistic absorption techniques (a method that pre-allocates a flexible reservoir of cells and then completes an almost-solution to an exact one), shows that for random-like equal-sized subsets $A,B,C$ of a sufficiently large group $G$ satisfying the natural sum condition $\sum A+\sum B=\sum C$ in $G^{\mathrm{ab}}$, there is a bijection $\phi:A\to B$ with $x\mapsto x\phi(x)$ a bijection from $A$ onto $C$. Among its consequences is the following classification, valid for all $n$ beyond some absolute constant $n_0$: the subsquare $A\times B$ has a transversal unless either (i) $A$ and $B$ are a left and a right coset of a subgroup $H$ failing the Hall–Paige condition (so $A\times B$ is a translate of $M(H)$, which has no transversal), or (ii) $A\times B$ arises from the multiplication table of some $(\mathbb{Z}_2)^k$ by deleting two rows and two columns whose entries sum to zero. (This is an informal description; the precise formalization of the two exceptional families, in which the deleted-index sets may be arbitrary left translates by elements $g_1,g_2\in G$, is given in the Problem Statement below.) This confirms both cases of Snevily's conjecture for sufficiently large subsquares, and as further consequences the paper characterises, for sufficiently large orders, the sequenceable groups (those whose elements admit an ordering with all partial products distinct), the $R$-sequenceable groups (the variant in which all partial products except the total are distinct and the total is the identity), and the harmonious groups (those whose elements admit an ordering in which the products of consecutive pairs are all distinct).

The proof is intrinsically asymptotic — the absorption and nibble arguments (the latter a randomized greedy process that covers almost all of a structure, leaving a small remainder for the absorber) need $n$ tending to infinity — and the authors state that it seems hopeless to push their methods down to small $n$. Nevertheless they conjecture that the classification itself is true for every $n$, and this is the problem audited here.

Problem Statement

Prove, or disprove, Conjecture 7.1 of Müyesser and Pokrovskiy: the following holds for every $n$ (with no lower bound on $n$). Let $G$ be a finite group and let $A,B\subseteq G$ with $|A|=|B|=n$. Then the subsquare $A\times B$ (rows indexed by $A$, columns by $B$, entry $ab$ in row $a$ and column $b$) contains a transversal — $n$ cells with no two sharing a row, a column, or a symbol — unless there exist an integer $k\ge 1$, elements $g_1,g_2\in G$, and a subgroup $H\subseteq G$ such that one of the following holds:

1. $H$ fails the Hall–Paige condition (equivalently, its Sylow 2-subgroups are nontrivial and cyclic), and $A=g_1H$ and $B=Hg_2$; or
2. $H\cong(\mathbb{Z}_2)^k$ and $g_1A=H\setminus\{a_1,a_2\}$, $g_2B=H\setminus\{b_1,b_2\}$ for distinct $a_1,a_2\in H$ and distinct $b_1,b_2\in H$ with $a_1+a_2+b_1+b_2=0$ (the sum computed in the abelian group $H$, written additively).

The quantitative strengthenings of the underlying matching theorem (the source paper's Conjecture 7.2 and Problems 7.3–7.4 on optimal error bounds), and the enumerative, strong-complete-mapping, and Kézdy–Snevily directions from the same concluding section, are neighboring open questions and are not part of the completion boundary of this problem.

References

1. Alp Müyesser and Alexey Pokrovskiy, A random Hall–Paige conjecture, Inventiones mathematicae 240:779–867 (2025), DOI 10.1007/s00222-025-01328-x; preprint arXiv:2204.09666, https://arxiv.org/abs/2204.09666
2. Marshall Hall and L. J. Paige, Complete mappings of finite groups, Pacific Journal of Mathematics 5:541–549 (1955), DOI 10.2140/pjm.1955.5.541, https://doi.org/10.2140/pjm.1955.5.541
3. H. S. Snevily, The Cayley addition table of Z_n, The American Mathematical Monthly 106(6):584–585 (1999), DOI 10.1080/00029890.1999.12005088, https://doi.org/10.1080/00029890.1999.12005088
4. Noga Alon, Additive Latin transversals, Israel Journal of Mathematics 117:125–130 (2000), DOI 10.1007/BF02773567, https://doi.org/10.1007/BF02773567
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8. Sean Eberhard, Freddie Manners, and Rudi Mrazović, An asymptotic for the Hall–Paige conjecture, Advances in Mathematics 404:108423 (2022), DOI 10.1016/j.aim.2022.108423; preprint arXiv:2003.01798, https://arxiv.org/abs/2003.01798
9. Richard Montgomery, Transversals in Latin squares, survey preprint arXiv:2406.19873 (2024), https://arxiv.org/abs/2406.19873
10. Alp Müyesser, Cycle type in Hall–Paige: a proof of the Friedlander–Gordon–Tannenbaum conjecture, Forum of Mathematics, Sigma (2026), DOI 10.1017/fms.2026.10197; preprint arXiv:2303.16157, https://arxiv.org/abs/2303.16157
11. Mohammad Javaheri, Symmetric harmoniousness of odd-order groups, preprint arXiv:2511.13430 (2025), https://arxiv.org/abs/2511.13430
12. Yutong Zhang and Yaoran Yang, The least non-partitionable zero-sum subset for zero-sum triples in finite abelian groups, preprint arXiv:2606.13764 (2026), https://arxiv.org/abs/2606.13764