Moufang loops with commuting inner mappings

TL;DR

This paper explores the structure of Moufang loops with commuting inner mappings, revealing conditions under which they have low nilpotency class and providing examples of loops with higher class.

Contribution

It establishes that Moufang loops of odd order or 6-divisible with commuting inner mappings have nilpotency class at most two, and constructs a specific example with class three.

Findings

Moufang loops of odd order with commuting inner mappings have nilpotency class ≤ 2

6-divisible Moufang loops with commuting inner mappings have nilpotency class ≤ 2

Existence of a Moufang loop of order 2^{14} with commuting inner mappings and nilpotency class 3

Abstract

We investigate the relation between the structure of a Moufang loop and its inner mapping group. Moufang loops of odd order with commuting inner mappings have nilpotency class at most two. $6$-divisible Moufang loops with commuting inner mappings have nilpotency class at most two. There is a Moufang loop of order $2^{14}$ with commuting inner mappings and of nilpotency class three.