Automorphic Loops

A loop (that is, a quasigroup with an identity element) is said to be automorphic if its inner mapping group acts as a group of automorphisms of the loop. Obviously groups are automorphic loops, but this class includes other important types of loops such as commutative Moufang loops. The study of automorphic loops has been a major theme in loop theory over the past decade. I spoke about this topic in CAUL about four years ago. I will discuss some recent developments, including the following. (1) We now have a very good understanding of finite commutative automorphic loops: they are all solvable and satisfy the Lagrange and Sylow/Hall (existence) Theorems. (It is rare for loops to satisfy the major theorems of group theory.) (2) The search for finite nonassociative simple automorphic loops continues. Such loops must have even order and their (necessarily primitive) multiplication groups cannot have regular socles, thus reducing the number of cases of the O’Nan-Scott classification which must be considered. What is perhaps surprising about both parts (1) and (2) are the key roles played by Lie algebras in the proofs.