Products of all elements in a loop and a framework for non-associative analogues of the Hall-Paige conjecture

TL;DR

This paper explores properties of finite loops related to the Hall-Paige conjecture, establishing new implications between conditions involving complete mappings, normal subloops, and element products, with generalizations and elementary proofs.

Contribution

It introduces a framework for non-associative analogues of the Hall-Paige conjecture, proving key implications and generalizations for finite loops.

Findings

Proves (A) implies (C) and (B) is equivalent to (C) in finite loops.

Establishes properties of the set of element products, P(Q).

Provides a generalized Dénes-Hermann theorem and an elementary proof of a weak Hall-Paige form.

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 $N \normal Q$ 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 D\'enes-Hermann theorem, and present an elementary proof of a weak form of the Hall-Paige conjecture.