Products of All Elements in a Loop and a Framework for Non-Associative Analogues of the Hall-Paige Conjecture

Abstract

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 $N \trianglelefteq Q$ such that $|N|$ is odd and $Q/N \cong {\Bbb Z}_{2^m}$ for $m \geq 1$, and
(C) $P(Q)$ intersects the associator subloop of $Q$.

We prove $(A) \Longrightarrow (C)$ and $(B) \Longleftrightarrow (C)$ 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 $(B) \Longrightarrow (A))$. 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.