We prove that the order of any subloop of a finite Moufang loop is a factor of the order of the loop, thus obtaining an analog of Lagrange's theorem for finite Moufang loops.

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Communicated by I. P. ShestakovA loop is said to be automorphic if its inner mappings are automorphisms. For a prime p, denote by Ap the class of all 2-generated commutative automorphic loops Q possessing a central subloop Z ∼ = Zp such that Q/Z ∼ = Zp × Zp. Upon describing the free 2-generated nilpotent class two commutative automorphic loop and the free 2-generated nilpotent class two commutative automorphic p-loop Fp in the variety of loops whose elements have order dividing p 2 and whose associators have order dividing p, we show that every loop of Ap is a quotient of Fp by a central subloop of order p 3 . The automorphism group of Fp induces an action of GL 2 (p) on the three-dimensional subspaces of Z(Fp) ∼ = (Zp) 4 . The orbits of this action are in one-to-one correspondence with the isomorphism classes of loops from Ap. We describe the orbits, and hence we classify the loops of Ap up to isomorphism.It is known that every commutative automorphic p-loop is nilpotent when p is odd, and that there is a unique commutative automorphic loop of order 8 with trivial center. Knowing Ap up to isomorphism, we easily obtain a classification of commutative automorphic loops of order p 3 . There are precisely seven commutative automorphic loops of order p 3 for every prime p, including the three abelian groups of order p 3 .