×

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.

MSC:

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

References:

[1] Drádal A., Ac ta Univ.Carolinae Math.Phys 34 pp 85– (1993)
[2] Drádal A., Acta Univ. Carolinae Math.Phys 35 pp 9– (1994)
[3] Jančařík A., Multiplication groups of quasigroups and loops III
[4] DOI: 10.1007/BF01198806 · Zbl 0789.20080 · doi:10.1007/BF01198806
[5] DOI: 10.1080/00927879608825558 · Zbl 0853.20049 · doi:10.1080/00927879608825558
[6] DOI: 10.1016/0021-8693(90)90152-E · Zbl 0706.20046 · doi:10.1016/0021-8693(90)90152-E
[7] DOI: 10.1017/S0004972700016166 · Zbl 0799.20020 · doi:10.1017/S0004972700016166
[8] Smith J.D.H., Multiplication groups of quasigroups 603 (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.