A random Hall-Paige conjecture
Alp Müyesser, Alexey Pokrovskiy · Inventiones mathematicae · 2025
Abstract A complete mapping of a group $G$ G is a bijection $\phi \colon G\to G$ ϕ : G → G such that $x\mapsto x\phi (x)$ x ↦ x ϕ ( x ) is also bijective. Hall and Paige conjectured in 1955 that a finite group $G$ G has a complete mapping whenever $\prod _{x\in G} x$ ∏ x ∈ G x is the identity in the abelianization of $G$ G . This was confirmed in 2009 by Wilcox, Evans, and Bray with a proof using the classification of finite simple groups. 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 $A$ A , $B$ B , $C$ C of a group $G$ G , there exists a bijection $\phi \colon A\to B$ ϕ : A → B such that $x\mapsto x\phi (x)$ x ↦ x ϕ ( x ) is a bijection from $A$ A to $C$ C whenever $\prod _{a\in A} a \prod _{b\in B} b=\prod _{c\in C} c$ ∏ a ∈ A a ∏ b ∈ B b = ∏ c ∈ C c in the abelianization of $G$ 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 $ π of their elements such that the partial products $\pi _{1}$ π 1 , $\pi _{1}\pi _{2}$ π 1 π 2 , $\pi _{1}\pi _{2}\cdots \pi _{n}$ π 1 π 2 ⋯ π