The Structure of Commutative Automorphic Loops

TL;DR

This paper investigates the structure of finite commutative automorphic loops, revealing their decomposition properties, element orders, and conditions for simplicity, with a focus on loops of prime power order and odd order.

Contribution

It provides a detailed structural analysis of finite commutative automorphic loops, including decomposition, divisibility, and simplicity criteria, advancing understanding of their algebraic properties.

Findings

Loops decompose into odd and 2-power order components.

Finite simple nonassociative commutative A-loops are of exponent 2.

If a prime divides the order, the loop contains an element of that order.

Abstract

An \emph{automorphic loop} (or \emph{A-loop}) is a loop whose inner mappings are automorphisms. Every element of a commutative A-loop generates a group, and $(xy)^{-1} = x^{-1}y^{-1}$ holds. Let $Q$ be a finite commutative A-loop and $p$ a prime. The loop $Q$ has order a power of $p$ if and only if every element of $Q$ has order a power of $p$. The loop $Q$ decomposes as a direct product of a loop of odd order and a loop of order a power of 2. If $Q$ is of odd order, it is solvable. If $A$ is a subloop of $Q$ then $|A|$ divides $|Q|$. If $p$ divides $|Q|$ then $Q$ contains an element of order $p$. If there is a finite simple nonassociative commutative A-loop, it is of exponent 2.