Searching for small simple automorphic loops

TL;DR

This paper investigates automorphic loops, establishing nonexistence of small nonassociative simple commutative automorphic loops below a certain order, and providing examples of nonassociative simple right automorphic loops.

Contribution

It proves bounds on the order of nonassociative simple automorphic loops and constructs new examples of nonassociative simple right automorphic loops.

Findings

No nonassociative simple commutative automorphic loops of order less than 4096.

No nonassociative simple automorphic loops of order less than 2500.

Existence of nonassociative simple right automorphic loops.

Abstract

A loop is (right) automorphic if all its (right) inner mappings are automorphisms. Using the classification of primitive groups of small degrees, we show that there is no nonassociative simple commutative automorphic loop of order less than $2^{12}$, and no nonassociative simple automorphic loop of order less that 2500. We obtain examples of nonassociative simple right automorphic loops.