On multiplication groups of loops.

*(English)*Zbl 0706.20046From the introduction: While studying the multiplication group of a loop \(Q\) a central role is played by the stabilizer of the neutral element. This subgroup \(I(Q)\) of the multiplication group is called the inner mapping group of \(Q\). It is known that a loop \(Q\) is an Abelian group if and only if \(I(Q)=1\). In this paper some properties of the inner mapping group are described. There is also given a partial answer to the question: What are the multiplication groups of loops? This is closely connected to certain transversal conditions. The multiplication groups of loops are characterized with the aid of these conditions. One of the main results is the theorem: If \(Q\) is a finite loop whose inner mapping group is cyclic then \(Q\) is an Abelian group. Finally it is shown that certain groups are not multiplication groups of loops.

Reviewer: M.Csikós

##### MSC:

20N05 | Loops, quasigroups |

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\textit{M. Niemenmaa} and \textit{T. Kepka}, J. Algebra 135, No. 1, 112--122 (1990; Zbl 0706.20046)

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