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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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##### References:
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