Conjugacy closed loops and their multiplication groups

A loop Q is said to be conjugacy closed if the sets {Lx; x Q} and {Rx; x Q} are closed under conjugation. Let and be the left and right multiplication groups of Q, respectively, and let InnQ be its inner mapping group. If Q is conjugacy closed, then there exist epimorphisms and that are determined by Lx R−1xLx and Rx L−1xRx. These epimorphisms are used to expose various structural properties of .