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

Kyle Pula

The electronic journal of combinatorics, Tome 16 (2009) no. 1

Résumé

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.

Détail

DOI : 10.37236/146

Classification : 20N05, 05B15
Mots-clés : Hall-Paige conjecture, finite loops, products of elements, complete mappings, multiplication tables