Constructions of commutative automorphic loops

TL;DR

This paper characterizes and constructs commutative automorphic loops, especially of order a power of 2, and initiates their classification for small orders and order p^3, advancing understanding of their structure.

Contribution

It provides a characterization of commutative automorphic loops with middle nucleus of index 2 and constructs new classes using central extensions, also initiating their classification for small orders.

Findings

Characterization of commutative automorphic loops with middle nucleus of index 2

Construction of new classes of commutative A-loops of order 2^n

Initiation of classification for small orders and order p^3

Abstract

A loop whose inner mappings are automorphisms is an \emph{automorphic loop} (or \emph{A-loop}). We characterize commutative (A-)loops with middle nucleus of index 2 and solve the isomorphism problem. Using this characterization and certain central extensions based on trilinear forms, we construct several classes of commutative A-loops of order a power of 2. We initiate the classification of commutative A-loops of small orders and also of order $p^3$, where $p$ is a prime.