Nilpotency in automorphic loops of prime power order

A loop is automorphic if its inner mappings are automorphisms. Using so-called associated operations, we show that every commutative automorphic loop of odd prime power order is centrally nilpotent. Starting with suitable elements of an anisotropic plane in the vector space of 2×2 matrices over the field of prime order p, we construct a family of automorphic loops of order p3 with trivial center.