### On the Summands of Near-Rings with ATM.

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We show that finite commutative inverse property loops with elementary abelian inner mapping groups of order ${p}^{4}$ are centrally nilpotent of class at most two.

We show that finite commutative inverse property loops may not have nonabelian dihedral 2-groups as their inner mapping group.

We investigate the situation that the inner mapping group of a loop is of order which is a product of two small prime numbers and we show that then the loop is soluble.

In this paper we consider finite loops of specific order and we show that certain abelian groups are not isomorphic to inner mapping groups of these loops. By using our results we are able to construct a finite solvable group of order 120 which is not isomorphic to the multiplication group of a finite loop.

In this paper we consider finite loops and discuss the following problem: Which groups are (are not) isomorphic to inner mapping groups of loops? We recall some known results on this problem and as a new result we show that direct products of dihedral 2-groups and nontrivial cyclic groups of odd order are not isomorphic to inner mapping groups of finite loops.

Loop capable groups are groups which are isomorphic to inner mapping groups of loops. In this paper we show that abelian groups ${C}_{p}^{k}\times {C}_{p}\times {C}_{p}$, where $k\ge 2$ and $p$ is an odd prime, are not loop capable groups. We also discuss generalizations of this result.

Let $Q$ be a loop such that $\left|Q\right|$ is square-free and the inner mapping group $I\left(Q\right)$ is nilpotent. We show that $Q$ is centrally nilpotent of class at most two.

Let $Q$ be a finite commutative loop and let the inner mapping group $I\left(Q\right)\cong {C}_{{p}^{n}}\times {C}_{{p}^{n}}$, where $p$ is an odd prime number and $n\ge 1$. We show that $Q$ is centrally nilpotent of class two.

Let $G$ be a finite group with a dicyclic subgroup $H$. We show that if there exist $H$-connected transversals in $G$, then $G$ is a solvable group. We apply this result to loop theory and show that if the inner mapping group $I\left(Q\right)$ of a finite loop $Q$ is dicyclic, then $Q$ is a solvable loop. We also discuss a more general solvability criterion in the case where $I\left(Q\right)$ is a certain type of a direct product.

We investigate the situation when the inner mapping group of a commutative loop is of order $2p$, where $p=4t+3$ is a prime number, and we show that then the loop is solvable.

In this paper we consider finite loops whose inner mapping groups are nilpotent. We first consider the case where the inner mapping group $I\left(Q\right)$ of a loop $Q$ is the direct product of a dihedral group of order $8$ and an abelian group. Our second result deals with the case where $Q$ is a $2$-loop and $I\left(Q\right)$ is a nilpotent group whose nonabelian Sylow subgroups satisfy a special condition. In both cases it turns out that $Q$ is centrally nilpotent.

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