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 $q^2$. 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.