Commutative automorphic loops of order $p^3$

A loop is said to be automorphic if its inner mappings are automorphisms. For a prime $p$, denote by $\mathcal A_p$ the class of all $2$-generated commutative automorphic loops $Q$ possessing a central subloop $Z\cong \mathbb Z_p$ such that $Q/Z\cong\mathbb Z_p\times\mathbb Z_p$. Upon describing the free $2$-generated nilpotent class two commutative automorphic loop and the free $2$-generated nilpotent class two commutative automorphic $p$-loop $F_p$ 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 $\mathcal A_p$ is a quotient of $F_p$ by a central subloop of order $p^3$. The automorphism group of $F_p$ induces an action of $GL_2(p)$ on the three-dimensional subspaces of $Z(F_p)\cong (\mathbb Z_p)^4$. The orbits of this action are in one-to-one correspondence with the isomorphism classes of loops from $\mathcal A_p$. We describe the orbits, and hence we classify the loops of $\mathcal A_p$ 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 $\mathcal A_p$ up to isomorphism, we easily obtain a classification of commutative automorphic loops of order $p^3$. There are precisely $7$ commutative automorphic loops of order $p^3$ for every prime $p$, including the $3$ abelian groups of order $p^3$.