On multiplication groups of left conjugacy closed loops

Abstract

summary: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 $\operatorname{Inn} Q$ be its inner mapping group. Then there exists a homomorphism \Cal L \to \operatorname{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