A random Hall-Paige conjecture

TL;DR

The paper develops a unified probabilistic-combinatorial framework to study complete mappings and related structures in finite groups, showing that a Hall–Paige-type obstruction in the abelianization is the only barrier for large groups when transferring to subset triple matchings. By recasting the problem as perfect matchings in a 3-uniform, 3-partite multiplication hypergraph and employing a robust absorption method, the authors derive a main theorem that applies to random-like, equal-sized subset triples $(X,Y,Z)$ with $ extstyle ig( extstyleigsum X+igsum Y+igsum Zig)=0$ in $G^{ ext{ab}}$. This approach yields strong consequences: near-complete characterizations of subsquares with transversals (Snevily-type results), the sequenceability and $R$-sequenceability of large groups, and comprehensive zero-sum partition results for large abelian groups, including resolutions of the Cichacz conjecture and Tannenbaum’s problem. Overall, the work provides a versatile, absorber-driven framework that unifies and extends several longstanding open problems at the interface of combinatorics and group theory, with potential for further counting, optimization, and refinement in dense algebraic structures.

Abstract

A complete mapping of a group $G$ is a bijection $φ\colon G\to G$ such that $x\mapsto xφ(x)$ is also bijective. Hall and Paige conjectured in 1955 that a finite group $G$ has a complete mapping whenever $\prod_{x\in G} x$ is the identity in the abelianization of $G$. This was confirmed in 2009 by Wilcox, Evans, and Bray with a proof using the classification of finite simple groups. \par In this paper, we give a combinatorial proof of a far-reaching generalisation of the Hall-Paige conjecture for large groups. We show that for random-like and equal-sized subsets $A,B,C$ of a group $G$, there exists a bijection $φ\colon A\to B$ such that $x\mapsto xφ(x)$ is a bijection from $A$ to $C$ whenever $\prod_{a\in A} a \prod_{b\in B} b=\prod_{c\in C} c$ in the abelianization of $G$. We use this statement as a black-box to settle the following old problems in combinatorial group theory for large groups. (1) We characterise sequenceable groups, that is, groups which admit a permutation $π$ of their elements such that the partial products $π_1$, $π_1π_2$, $π_1π_2\cdots π_n$ are all distinct. This resolves a problem of Gordon from 1961 and confirms conjectures made by several authors, including Keedwell's 1981 conjecture that all large non-abelian groups are sequenceable. We also characterise the related $R$-sequenceable groups, addressing a problem of Ringel from 1974. (2) We confirm in a strong form a conjecture of Snevily from 1999 by characterising large subsquares of multiplication tables of finite groups that admit transversals. Previously, this characterisation was known only for abelian groups of odd order (by a combination of papers by Alon and Dasgupta-Károlyi-Serra-Szegedy and Arsovski).

Figures (12)

• Figure 1: The gadget $Q_x$. Matchings $M_1$ and $M_2$ depicted in solid and dashed lines, respectively.
• Figure 2: The set $S\subseteq G\ast F_3$ in Lemma \ref{['Lemma_absorber_one_commutator']}. Black letters $x,y,z$ are free variables, while pink letters are elements of $G$. The two elements $[a,b]c$, $c$ are not part of $S$ (and are just pictured to show how $S$$1$-absorbs $\{[a,b]c,c\}$).
• Figure 3: Proofs for weak/strong separability and linearity of all pairs $w,w'\in S$ in Lemma \ref{['Lemma_absorber_one_commutator']} to justify the application of Lemma \ref{['Lemma_separated_set_random']}. For strong separability, first we have partitioned $S$ into five subsets $S=S_{x}\cup S_{x,y}\cup S_{x,z}\cup S_{z}\cup S_{y}$ based on which free variables appear in each $w\in S$ (as in Observation \ref{['Observation_partition_S_by_free_variables']}). By Observation \ref{['Observation_partition_S_by_free_variables']}, any $w,w'$ in different subsets are strongly separable by part (a) of the definition. For each of the sets $S_{x}, S_{x,y}, S_{x,z}, S_{z}, S_{y}$ we give a table explaining why the $w,w'$ in that set are strongly/weakly-separable. Note that for words coming from different $S_A/S_B/S_C$ we need to show strong separability, but for words coming from the same part, weak separability suffices. Blue cells represent $w,w'$ being strongly separable via part (b) of the definition, green cells represent $w,w'$ being weakly separable via part (b'), and grey cells represent $w,w'$ not being separable/weakly-separable. The group element inside each blue cell is a generic element $g$ so that $w'\in\{gw, g^{-1}w, gw^{-1}, g^{-1}w^{-1}, wg, wg^{-1}, w^{-1}g, w^{-1}g^{-1}\}$ (thus checking (b) for $w,w'$). The group element inside the green cells is a non-identity element $g$ so that $w=w'$ rearranges into ${e}=g$ (thus checking (b') for $w,w'$). Observe that green cells are used only between pairs of words coming from the same part of $S_A/S_B/S_C$, meaning that we have strong separation for pairs of words coming from different parts $S_A/S_B/S_C$, as needed. To see that every $s\in S$ is linear notice that every word pictured has no repetitions of black letters.
• Figure 4: The set of words $S\subseteq G\ast F_2$ for Lemma \ref{['Lemma_absorber_3set']}. First, second, and third row of words represent $S_A$, $S_B$, and $S_C$, respectively. Black letters $u,w$ are free variables of $F_2$, while pink letters are elements of $G$. The elements $a,b,c,d, bwua, dwuc$ are not part of $S$ (and are pictured just to demonstrate how the absorption works).
• Figure 5: Justification for linearity and separability of pairs $(w,w')$ in $S$ in Lemma \ref{['Lemma_absorber_3set']}. For separability, first split $S$ into $S_u=\{u, { a^{-1}}u^{-1}, { c^{-1}}u^{-1}\}$, $S_w=\{w, { b^{-1}}w^{-1}, { d^{-1}}w^{-1}\}$ based on which free variables appear in the elements (as in Observation \ref{['Observation_partition_S_by_free_variables']}). By Observation \ref{['Observation_partition_S_by_free_variables']}, pairs $(w,w')$ with $w,w'$ in different sets $S_u/S_w$ fall under part (a) of the definition of strongly separable, so it remains to check pairs inside $S_u$ and $S_w$. Justification for this is given in the two tables above (with the same conventions as in Figure \ref{['Figure_justification_commutator']}, in particular, green cells are used only between pairs of vertices coming from the same part $S_A/S_B/S_C$). Relevant elements are generic/non-identity as a consequence of $a,b,c,d$ being distinct and generic). To see that each $w\in S$ is linear, note that there are no repetitions of black letters in each $w$ (and each $w\in S$ contains at least one black letter).

Theorems & Definitions (223)

• Theorem 1.1: Main result
• Proposition 1.2
• proof
• Conjecture 1.3: Snevily, snevily
• Theorem 1.4
• Theorem 1.5
• Conjecture 1.6: Cichacz, cichaczconjecture
• Proposition 2.1
• Lemma 2.2: Simpler version of Lemma \ref{['lem:zerosumabsorptionnonabelian']}
• Lemma 2.3: Simpler version of Lemma \ref{['lem:cosetpairedabsorber']}