Abstract:A loop $Q$ is said to be left conjugacy closed (LCC) if the set $\{L_x; x \in Q\}$ is closed under conjugation. Let $Q$ be such a loop, let $\Cal L$ and $\Cal R$ be the left and right multiplication groups of $Q$, respectively, and let $Inn Q$ be its inner mapping group. Then there exists a homomorphism $\Cal L \to Inn Q$ determined by $L_x \mapsto R^{-1}_xL_x$, and the orbits of $[\Cal L, \Cal R]$ coincide with the cosets of $A(Q)$, the associator subloop of $Q$. All LCC loops of prime order are abelian groups.
Keywords: left conjugacy closed loop, multiplication group, nucleus
AMS Subject Classification: Primary 20N05; Secondary 08A05