On the Summands of Near-Rings with ATM.
On Near-Rings with ATM.
On finite commutative IP-loops with elementary abelian inner mapping groups of order
We show that finite commutative inverse property loops with elementary abelian inner mapping groups of order are centrally nilpotent of class at most two.
A note on systems of ideals
On finite loops and their inner mapping groups
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.
On the structure of finite loop capable Abelian groups
Loop capable groups are groups which are isomorphic to inner mapping groups of loops. In this paper we show that abelian groups , where and is an odd prime, are not loop capable groups. We also discuss generalizations of this result.
On dihedral 2-groups as inner mapping groups of finite commutative inverse property loops
We show that finite commutative inverse property loops may not have nonabelian dihedral 2-groups as their inner mapping group.
On abelian inner mapping groups of finite loops
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.
On finite loops whose inner mapping groups have small orders
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.
On finite commutative IP-loops with elementary abelian inner mapping groups of order
We show that finite commutative inverse property loops with elementary abelian inner mapping groups of order are centrally nilpotent of class at most two.
On a General Associativity Law in Groupoids.
On finite commutative loops which are centrally nilpotent
Let be a finite commutative loop and let the inner mapping group , where is an odd prime number and . We show that is centrally nilpotent of class two.
On splitting -systems
Normal subgroups as ideals
On central nilpotency in finite loops with nilpotent inner mapping groups
In this paper we consider finite loops whose inner mapping groups are nilpotent. We first consider the case where the inner mapping group of a loop is the direct product of a dihedral group of order and an abelian group. Our second result deals with the case where is a -loop and is a nilpotent group whose nonabelian Sylow subgroups satisfy a special condition. In both cases it turns out that is centrally nilpotent.
A note on loops of square-free order
Let be a loop such that is square-free and the inner mapping group is nilpotent. We show that is centrally nilpotent of class at most two.
On dicyclic groups as inner mapping groups of finite loops
Let be a finite group with a dicyclic subgroup . We show that if there exist -connected transversals in , then is a solvable group. We apply this result to loop theory and show that if the inner mapping group of a finite loop is dicyclic, then is a solvable loop. We also discuss a more general solvability criterion in the case where is a certain type of a direct product.
On the solvability of commutative loops and their multiplication groups
We investigate the situation when the inner mapping group of a commutative loop is of order , where is a prime number, and we show that then the loop is solvable.
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