  An \emph{automorphic loop} (or \emph{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.
