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On the Abelian inner permutation groups of loops. (English) Zbl 0913.20043
The main theorem of this paper is: Let \(Q\) be a loop such that the inner permutation group \(I(Q)\) of \(Q\) is finite and Abelian. Then \(Q\) is centrally nilpotent and no non-trivial primary \(p\)-component of \(I(Q)\) is cyclic.
This result is a generalization of previous works of M. Niemenmaa [Commun. Algebra 24, No. 1, 135-142 (1996; Zbl 0853.20049)] and M. Niemenmaa and T. Kepka [Bull. Aust. Math. Soc. 49, No. 1, 121-128 (1994; Zbl 0799.20020)], where a restrictive assumption in addition was that \(Q\) is a finite loop.

20N05 Loops, quasigroups
20B35 Subgroups of symmetric groups
Full Text: DOI
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