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conjugacy closed loops; Buchsteiner loops
We give sufficient and in some cases necessary conditions for the conjugacy closedness of $Q/Z(Q)$ provided the commutativity of $Q/N$. We show that if for some loop $Q$, $Q/N$ and $\operatorname{Inn} Q$ are abelian groups, then $Q/Z(Q)$ is a CC loop, consequently $Q$ has nilpotency class at most three. We give additionally some reasonable conditions which imply the nilpotency of the multiplication group of class at most three. We describe the structure of Buchsteiner loops with abelian inner mapping groups.
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