The structure of automorphic loops

TL;DR

This paper investigates the structure and properties of automorphic loops, especially those of odd order, establishing their solvability, subgroup divisibility, and the nonexistence of certain simple loops, while constructing specific nonassociative examples.

Contribution

It provides new results on the structure, solvability, and classification of automorphic loops, including the construction of nonassociative automorphic loops of certain orders and the proof of nonexistence of simple nonassociative automorphic loops below order 2500.

Findings

Automorphic loops of odd order are solvable.

No finite simple nonassociative automorphic loops of order less than 2500.

Existence of nonassociative automorphic loops of order p^3 and dihedral loops of order 2n.

Abstract

Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes, for instance, groups and commutative Moufang loops. We study uniquely 2-divisible automorphic loops, particularly automorphic loops of odd order, from the point of view of the associated Bruck loops (motivated by Glauberman's work on uniquely 2-divisible Moufang loops) and the associated Lie rings (motivated by a construction of Wright). We prove that every automorphic loop $Q$ of odd order is solvable, contains an element of order $p$ for every prime $p$ dividing $|Q|$, and $|S|$ divides $|Q|$ for every subloop $S$ of $Q$. There are no finite simple nonassociative commutative automorphic loops, and there are no finite simple nonassociative automorphic loops of order less than 2500. We show that if $Q$ is a finite simple nonassociative automorphic loop then the socle of the multiplication group of $Q$ is not regular. The existence of a finite simple nonassociative automorphic loop remains open. Let $p$ be an odd prime. Automorphic loops of order $p$ or $p^2$ are groups, but there exist nonassociative automorphic loops of order $p^3$, some with trivial nucleus (center) and of exponent $p$. We construct nonassociative "dihedral" automorphic loops of order $2n$ for every $n>2$, and show that there are precisely $p-2$ nonassociative automorphic loops of order $2p$, all of them dihedral.