The Structure of Conjugacy Closed Loops

We study structure theorems for the conjugacy closed (CC-) loops, a specific variety of G-loops (loops isomorphic to all their loop isotopes). These theorems give a description all such loops of small order. For example, if p and q are primes, p < q, and q − 1 is not divisible by p, then the only CC-loop of order pq is the cyclic group of order pq. For any prime q > 2, there is exactly one non-group CC-loop in order 2q, and there are exactly three in order q2. We also derive a number of equations valid in all CC-loops. By contrast, every equation valid in all G-loops is valid in all loops.