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On abelian inner mapping groups of finite loops

Markku Niemenmaa — 2000

Commentationes Mathematicae Universitatis Carolinae

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 and their inner mapping groups

Markku Niemenmaa — 2004

Commentationes Mathematicae Universitatis Carolinae

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

Markku Niemenmaa — 2007

Commentationes Mathematicae Universitatis Carolinae

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 × C p × C p , where k 2 and p is an odd prime, are not loop capable groups. We also discuss generalizations of this result.

On dicyclic groups as inner mapping groups of finite loops

Emma LeppäläMarkku Niemenmaa — 2016

Commentationes Mathematicae Universitatis Carolinae

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 ( Q ) 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 ( Q ) is a certain type of a direct product.

On central nilpotency in finite loops with nilpotent inner mapping groups

Markku NiemenmaaMiikka Rytty — 2008

Commentationes Mathematicae Universitatis Carolinae

In this paper we consider finite loops whose inner mapping groups are nilpotent. We first consider the case where the inner mapping group I ( Q ) 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 ( Q ) 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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