Power-associative, conjugacy closed loops
  We study conjugacy closed loops (CC-loops) and power-associative CC-loops (PACC-loops). If $Q$ is a PACC-loop with nucleus $N$, then $Q/N$ is an abelian group of exponent 12; if in addition $Q$ is finite, then $|Q|$ is divisible by 16 or by 27. There are eight nonassociative PACC-loops of order 16, three of which are not extra loops. There are eight nonassociative PACC-loops of order 27, four of which have the automorphic inverse property.   We also study some special elements in loops, such as Moufang elements, weak inverse property (WIP) elements, and extra elements. In a CC-loop, the set of WIP and the set of extra elements are normal subloops. For each $c$ in a PACC-loop, $c^3$ is WIP, $c^6$ is extra, and $c^{12} \in N$. 
Michael K. Kinyon and Kenneth Kunen