Three lectures on automorphic loops

Abstract

These notes accompany a series of three lectures on automorphic loops to be delivered by the author at Workshops Loops '15 (Ohrid, Macedonia, 2015). Automorphic loops are loops in which all inner mappings are automorphisms. The first paper on automorphic loops appeared in 1956 and there has been a surge of interest in the topic since 2010. The purpose of these notes is to introduce the methods used in the study of automorphic loops to a wider audience of researchers working in nonassociative mathematics. In the first lecture we establish basic properties of automorphic loops (flexibility, power-associativity and the antiautomorphic inverse property) and discuss relations of automorphic loops to Moufang loops. In the second lecture we expand on ideas of Glauberman and investigate the associated operation $(x^{-1}\backslash (y^2x))^{1/2}$ and similar concepts, using a more modern approach of twisted subgroups. We establish many structural results for commutative and general automorphic loops, including the Odd Order Theorem. In the last lecture we look at enumeration and constructions of automorphic loops. We show that there are no nonassociative simple automorphic loops of order less than $4096$, we study commutative automorphic loops of order $pq$ and $p^3$, and introduce two general constructions of automorphic loops. The material is newly organized and sometimes new, shorter proofs are given.