Abelian congruences and solvability in Moufang loops

TL;DR

This paper investigates the relationship between abelian normal subloops and congruences in Moufang loops, showing that in certain cases they coincide and establishing a strong form of solvability for odd order Moufang loops.

Contribution

It constructs examples of Moufang loops where abelian normal subloops do not induce abelian congruences and proves the equivalence of solvability notions in specific classes of Moufang loops.

Findings

Centrally nilpotent Moufang loops can have abelian normal subloops that do not induce abelian congruences.

In 6-divisible Moufang loops, every abelian normal subloop induces an abelian congruence.

All Moufang loops of odd order are congruence solvable, extending Glauberman's Odd Order Theorem.

Abstract

In groups, an abelian normal subgroup induces an abelian congruence. We construct a class of centrally nilpotent Moufang loops containing an abelian normal subloop that does not induce an abelian congruence. On the other hand, we prove that in $6$-divisible Moufang loops, every abelian normal subloop induces an abelian congruence. In loops, congruence solvability adopted from the universal-algebraic commutator theory of congruence modular varieties is strictly stronger than classical solvability adopted from group theory. It is an open problem whether the two notions of solvability coincide in Moufang loops. We prove that they coincide in $6$-divisible Moufang loops and in Moufang loops of odd order. In fact, we show that every Moufang loop of odd order is congruence solvable, thus strengthening Glauberman's Odd Order Theorem for Moufang loops.