The structure of commutative automorphic loops

Abstract

An automorphic loop (or 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 $2$.