论文标题
一个随机的霍尔·塔吉（Hall-Paige）猜想
A random Hall-Paige conjecture
论文作者
论文摘要
组$ g $的完整映射是Biokey $ ϕ \ colon g \ to g $，因此$ x \ mapsto xϕ（x）$也是生物。 Hall and Paige在1955年猜想，只要$ \ prod_ {x \ in G} x $，有限的组$ g $都是$ g $的abelianization的身份。 Wilcox，Evans和Bray在2009年证实了这一点，并使用有限简单组的分类进行了证明。 \在本文中，我们给出了组合证据，证明了大型群体对Hall-Paige猜想的深远概括。我们表明，对于一组$ g $的随机且相等大小的子集$ a，b，c $，存在一个两者的两者，只要$ x \ mapsto x \ x \ x \ x x x（x）$是$ a $ a $ a $ a $ a $ a to $ c $ b的$ c $，每当$ \ prod_ in} a \ in} a \ in} a} a \ frot_ b} $ g $的Abelianization中的C $。我们将此陈述用作黑框，以解决大型组的组合群体理论中的以下旧问题。 （1）我们表征可测序的群体，即接纳元素的排列$π$的组，使得部分产品$π_1$，$π_1π_2$，$π_1π_2\cdotsπ_n$都是不同的。这从1961年开始解决了戈登的问题，并确认了几位作者的猜想，包括基德威尔（Keedwell）1981年的猜想，即所有大型非亚伯群都可以进行测序。我们还表征了相关的$ r $序列组，解决了1974年林德尔的问题。（2）我们以强烈的形式确认了1999年从1999年开始的猜想，通过表征大型有限群体的乘法表的大量子等，这些乘以横向横向。以前，这种特征仅以奇数秩序的亚伯群（通过Alon和Dasgupta-károlyi-Serra-Sera-Szegedy和Arsovski的论文组合）而闻名。
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).
