Congruence solvability in finite Moufang loops of order coprime to three

TL;DR

This paper characterizes when a normal subloop in finite Moufang loops of order coprime to three induces an abelian congruence, and applies this to show classical solvability implies congruence solvability in such loops.

Contribution

It provides a new characterization of abelian congruences in finite Moufang loops of order coprime to three, linking inner mappings and subloop properties.

Findings

Normal subloop induces abelian congruence iff inner mappings restrict to automorphisms and a specific identity holds.

In 3-divisible Moufang loops, the automorphism restriction condition can be omitted.

Classically solvable finite 3-divisible Moufang loops are congruence solvable.

Abstract

We prove that a normal subloop $X$ of a Moufang loop $Q$ induces an abelian congruence of $Q$ if and only if each inner mapping of $Q$ restricts to an automorphism of $X$ and $u(xy) = (uy)x$ for all $x,y\in X$ and $u\in Q$. The former condition can be omitted when $X$ is $3$-divisible. This characterization is then used to show that classically solvable finite $3$-divisible Moufang loops are congruence solvable.