COMMUTATIVE AUTOMORPHIC LOOPS OF ORDER p3

Abstract

A loop is said to be automorphic if its inner mappings are automorphisms. For a prime p, denote by [Formula: see text] 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 p2 and whose associators have order dividing p, we show that every loop of [Formula: see text] is a quotient of Fp by a central subloop of order p3. 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 [Formula: see text]. We describe the orbits, and hence we classify the loops of [Formula: see text] 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 [Formula: see text] up to isomorphism, we easily obtain a classification of commutative automorphic loops of order p3. There are precisely seven commutative automorphic loops of order p3 for every prime p, including the three abelian groups of order p3.