A random Hall-Paige conjecture

• 成果类型: Article
• 署名作者: Muyesser, Alp; Pokrovskiy, Alexey
• 署名单位: University of Oxford; University of London; University College London
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01328-x
• 发表日期: 2025
• 页码: 779-867
• 关键词: transversals; partitions

摘要

A complete mapping of a group G is a bijection phi:G -> G such that x bar right arrow x phi(x) is also bijective. Hall and Paige conjectured in 1955 that a finite group G has a complete mapping whenever Pi(x is an element of 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. 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:A -> B such that x bar right arrow x phi(x) is a bijection from A to C whenever Pi(a is an element of A)a Pi(b is an element of B)b=Pi;(c is an element of 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). . .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-Karolyi-Serra-Szegedy and Arsovski). (3) We characterise the abelian groups that can be partitioned into zero-sum sets of specific sizes, solving a problem of Tannenbaum from 1981. This also confirms a recent conjecture of Cichacz. (4) We characterise harmonious groups, that is, groups with an ordering in which the product of each consecutive pair of elements is distinct, solving a problem of Evans from 2015.