A random Hall-Paige conjecture

Abstract

A complete mapping of a group $G$ is a bijection $\phi\colon G\to G$ such that $x\mapsto x\phi(x)$ 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 $\phi\colon A\to B$ such that $x\mapsto x\phi(x)$ 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 $\pi$ of their elements such that the partial products $\pi_1$, $\pi_1\pi_2$, $\pi_1\pi_2\cdots \pi_n$ 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-K\'arolyi-Serra-Szegedy and Arsovski).