The structure of commutative automorphic loops

Transactions of the American Mathematical Society, 363(1) (2011), 365-384
http://www.ams.org/journals/tran/2011-363-01/S0002-9947-2010-05088-3/

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 holds. Let be a finite commutative A-loop and a prime. The loop has order a power of if and only if every element of has order a power of . The loop decomposes as a direct product of a loop of odd order and a loop of order a power of . If is of odd order, it is solvable. If is a subloop of , then divides . If divides , then contains an element of order . If there is a finite simple nonassociative commutative A-loop, it is of exponent..