循环中所有元素的乘积和Hall-Paige猜想的非结合类似物的框架
Products of all elements in a loop and a framework for non-associative analogues of the Hall-Paige conjecture
摘要：
对于有限循环$Q$，令$P (Q)$是可以表示为包含$Q$的每个元素的乘积的元素的集合。受Hall-Paige猜想的最近证明的启发，我们证明了下列条件之间的几个普遍含义：（A）$Q$有一个完全映射，即$Q$的乘法表有一个横截;（B）不存在$N \normal Q$使得$|N|$是奇数，且$Q/N \cong \ZZ_{2^m}$对于$m \geq 1$;（C）$P(Q)$与$Q$的结合子环相交.我们证明$(A) \implies (C)$和$(B) \iff (C)$，并表明，当$Q$是一个群体，这些条件减少到熟悉的陈述有关的霍尔-佩奇猜想（这基本上说，在群体$(B) \implies (A))$。我们还建立了$P(Q)$的性质，证明了Dénes-Hermann定理的推广，并给出了Hall-Paige猜想的一个弱形式的初等证明。
英文摘要：
For a finite loop $Q$, let $P (Q)$ be the set of elements that can be represented as a product containing each element of $Q$ precisely once. Motivated by the recent proof of the Hall-Paige conjecture, we prove several universal implications between the following conditions: (A) $Q$ has a complete mapping, i.e. the multiplication table of $Q$ has a transversal, (B) there is no $N \normal Q$ such that $|N|$ is odd and $Q/N \cong \ZZ_{2^m}$ for $m \geq 1$, and (C) $P(Q)$ intersects the associator subloop of $Q$. We prove $(A) \implies (C)$ and $(B) \iff (C)$ and show that when $Q$ is a group, these conditions reduce to familiar statements related to the Hall-Paige conjecture (which essentially says that in groups $(B) \implies (A))$. We also establish properties of $P(Q)$, prove a generalization of the Dénes-Hermann theorem, and present an elementary proof of a weak form of the Hall-Paige conjecture.
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