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Keywords:
left conjugacy closed loop; multiplication group; nucleus
left conjugacy closed loop; multiplication group; nucleus
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.
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