迭代群和拟群中的横截、近横截和对角线
Transversals, near transversals, and diagonals in iterated groups and quasigroups
摘要：
给定一个$n$阶的二元拟群$G$，一个$d$ -迭代拟群$G[d]$是等于$G$与自身的$d$ -乘合成的$(d+1)$ -元拟群.每一个$d$元拟群的Cayley表都是一个$d$维拉丁超立方体.定义了多元拟群中的横截和对角，使之与相应的拉丁超立方体中的横截和对角一致。证明了若$n$阶群$G$满足Hall-Paige条件，则对于大的$d$，$G[d]$中的断面数等于$ \frac{n!}{ |G'| n^{n-1}} \cdot n!^{d} (1 + o(1))$，其中$G'$是$G$的换位子群.对于一般拟群$G$，我们得到了类似的关于$G[d]$中断面和近断面个数的估计，并发展了一种计算迭代拟群中其它类型对角线的方法.
英文摘要：
Given a binary quasigroup $G$ of order $n$, a $d$-iterated quasigroup $G[d]$ is the $(d+1)$-ary quasigroup equal to the $d$-times composition of $G$ with itself. The Cayley table of every $d$-ary quasigroup is a $d$-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so as to coincide with those in the corresponding latin hypercube. We prove that if a group $G$ of order $n$ satisfies the Hall--Paige condition, then the number of transversals in $G[d]$ is equal to $ \frac{n!}{ |G'| n^{n-1}} \cdot n!^{d} (1 + o(1))$ for large $d$, where $G'$ is the commutator subgroup of $G$. For a general quasigroup $G$, we obtain similar estimations on the numbers of transversals and near transversals in $G[d]$ and develop a method for counting diagonals of other types in iterated quasigroups.
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