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The structure of commutative automorphic loops

Jedlicka, P.; Kinyon, M.; Vojtechovsky, P.

Transactions of the American Mathematical Society, 363(1) (2011), 365-384

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..