Structural interactions of conjugacy closed loops

Abstract:

We study conjugacy closed loops by means of their multiplication groups. Let $Q$ be a conjugacy closed loop, $N$ its nucleus, $A$ the associator subloop, and $\mathcal L$ and $\mathcal R$ the left and right multiplication groups, respectively. Put $M = \{a\in Q;$ $L_a \in \mathcal R\}$. We prove that the cosets of $A$ agree with orbits of $[\mathcal L, \mathcal R]$, that $Q/M \cong (\operatorname {Inn} Q)/\mathcal L_1$ and that one can define an abelian group on $Q/N \times \operatorname {Mlt}_1$. We also explain why the study of finite conjugacy closed loops can be restricted to the case of $N/A$ nilpotent. Group $[\mathcal {L},\mathcal {R}]$ is shown to be a subgroup of a power of $A$ (which is abelian), and we prove that $Q/N$ can be embedded into $\operatorname {Aut}([\mathcal {L}, \mathcal {R}])$. Finally, we describe all conjugacy closed loops of order $pq$.