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We give sufficient and in some cases necessary conditions for the conjugacy closedness of $Q/Z\left(Q\right)$ provided the commutativity of $Q/N$. We show that if for some loop $Q$, $Q/N$ and $InnQ$ are abelian groups, then $Q/Z\left(Q\right)$ 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.

A loop $Q$ is said to be left conjugacy closed if the set $A=\{L_x/x\in Q\}$ is closed under conjugation. Let $Q$ be an LCC loop, let $ℒ$ and $ℛ$ be the left and right multiplication groups of $Q$ respectively, and let $I\left(Q\right)$ be its inner mapping group, $M\left(Q\right)$ its multiplication group. By Drápal’s theorem [3, Theorem 2.8] there exists a homomorphism $\Lambda :ℒ\to I\left(Q\right)$ determined by $L_x\to R_x^{-1}{L}_x$. In this short note we examine different possible extensions of this $\Lambda$ and the uniqueness of these extensions.

Multiplication groups of (finite) loops with commuting inner permutations are investigated. Special attention is paid to the normal closure of the abelian permutation group.