A random Hall-Paige conjecture

A complete mapping of a group G G is a bijection φ G→ G ϕ : G → G such that x↦ xφ (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 ∏ x∈ 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 φ A→ B ϕ : A → B such that x↦ xφ (x) x ↦ x ϕ ( x ) is a bijection from A A to C C whenever ∏ a∈ A a ∏ b∈ B b=∏ c∈ 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 π π of their elements such that the partial products π ₁ π 1 , π ₁π ₂ π 1 π 2 , π ₁π ₂⋯ π ₙ π 1 π 2 ⋯ π