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On the structure of the inner mapping groups of loops. (English) Zbl 0853.20049
Let \(Q\) be a loop and \(M(Q)\) be the multiplication group of \(Q\), i.e. the subgroup of the symmetric group \(S(Q)\) generated by the set of all left and right translations. If \(I(Q)\) is a stabilizer of the neutral element of the loop \(Q\), then we say that \(I(Q)\) is the inner mapping group of \(Q\), as well. In previous papers due to T. Kepka, M. Niemenmaa and G. Rosenberger it was shown that if \(I(Q)\) is cyclic, then \(Q\) is abelian group and, consequently, \(I(Q)=1\). In the paper under review, this result is generalized for the case of finite loops by proving that in this case \(I(Q)\) cannot be isomorphic to a direct product \(C\times D\), where \(C\) is a nontrivial cyclic group, \(D\) is an abelian group, and \((|C|,|D|)=1\).

MSC:
20N05 Loops, quasigroups
20B35 Subgroups of symmetric groups
20F29 Representations of groups as automorphism groups of algebraic systems
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