Inverse property of nonassociative abelian extensions

Ágota Figula; Péter T. Nagy

Commentationes Mathematicae Universitatis Carolinae (2020)

  • Volume: 61, Issue: 4, page 501-511
  • ISSN: 0010-2628

Abstract

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Our paper deals with the investigation of extensions of commutative groups by loops so that the quasigroups that result in the multiplication between cosets of the kernel subgroup are T-quasigroups. We limit our study to extensions in which the quasigroups determining the multiplication are linear functions without constant term, called linear abelian extensions. We characterize constructively such extensions with left-, right-, or inverse properties using a general construction according to an equivariant group action principle. We show that the obtained constructions can be simplified for ordered loops. Finally, we apply our characterization to determine the possible cardinalities of the component loop of finite linear abelian extensions.

How to cite

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Figula, Ágota, and Nagy, Péter T.. "Inverse property of nonassociative abelian extensions." Commentationes Mathematicae Universitatis Carolinae 61.4 (2020): 501-511. <http://eudml.org/doc/297165>.

@article{Figula2020,
abstract = {Our paper deals with the investigation of extensions of commutative groups by loops so that the quasigroups that result in the multiplication between cosets of the kernel subgroup are T-quasigroups. We limit our study to extensions in which the quasigroups determining the multiplication are linear functions without constant term, called linear abelian extensions. We characterize constructively such extensions with left-, right-, or inverse properties using a general construction according to an equivariant group action principle. We show that the obtained constructions can be simplified for ordered loops. Finally, we apply our characterization to determine the possible cardinalities of the component loop of finite linear abelian extensions.},
author = {Figula, Ágota, Nagy, Péter T.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {loop; nonassociative extensions of abelian groups; linear abelian extensions; left property; right property; inverse property},
language = {eng},
number = {4},
pages = {501-511},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Inverse property of nonassociative abelian extensions},
url = {http://eudml.org/doc/297165},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Figula, Ágota
AU - Nagy, Péter T.
TI - Inverse property of nonassociative abelian extensions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 4
SP - 501
EP - 511
AB - Our paper deals with the investigation of extensions of commutative groups by loops so that the quasigroups that result in the multiplication between cosets of the kernel subgroup are T-quasigroups. We limit our study to extensions in which the quasigroups determining the multiplication are linear functions without constant term, called linear abelian extensions. We characterize constructively such extensions with left-, right-, or inverse properties using a general construction according to an equivariant group action principle. We show that the obtained constructions can be simplified for ordered loops. Finally, we apply our characterization to determine the possible cardinalities of the component loop of finite linear abelian extensions.
LA - eng
KW - loop; nonassociative extensions of abelian groups; linear abelian extensions; left property; right property; inverse property
UR - http://eudml.org/doc/297165
ER -

References

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  1. Albert A. A., 10.1090/S0002-9947-1944-0010597-1, Trans. Amer. Math. Soc. 55 (1944), 401–419. MR0010597DOI10.1090/S0002-9947-1944-0010597-1
  2. Bruck R. H., 10.1090/S0002-9947-1944-0011083-5, Trans. Amer. Math. Soc. 56 (1944), 141–199. MR0011083DOI10.1090/S0002-9947-1944-0011083-5
  3. Kalhoff F. B., Prieß-Krampe S. H. G., Ordered loops and ordered planar ternary rings, Chap. XIV in Quasigroups and Loops: Theory and Applications, Sigma Ser. Pure Math., 8, Heldermann, Berlin, 1990, pages 445–466. MR1125820
  4. Nagy P. T., Strambach K., 10.1007/s10587-008-0050-7, Czechoslovak Math. J. 58 (2008), no. 3, 759–786. Zbl1166.20058MR2455937DOI10.1007/s10587-008-0050-7
  5. Němec P., Kepka T., T-quasigroups. I, Acta Univ. Carolin. Math. Phys. 12 (1971), no. 1, 39–49. MR0320206
  6. Němec P., Kepka T., T-quasigroups. II, Acta Univ. Carolin. Math. Phys. 12 (1972), no. 2, 31–49. MR0320206
  7. Pflugfelder H. O., Quasigroups and Loops: Introduction, Sigma Series in Pure Mathematics, 7, Heldermann Verlag, Berlin, 1990. Zbl0715.20043MR1125767
  8. Shcherbacov V., Elements of quasigroup theory and applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, 2017. MR3644366
  9. Stanovský D., Vojtěchovský P., 10.1007/s00025-014-0382-6, Results Math. 66 (2014), no. 3–4, 367–384. MR3272634DOI10.1007/s00025-014-0382-6
  10. Stanovský D., Vojtěchovský P., Central and medial quasigroups of small order, Bul. Acad. Ştiinţe Repub. Mold. Mat. 80 (2016), no. 1, 24–40. Zbl1349.20075MR3528005
  11. Suvorov N. M., Kryuchkov N. I., 10.1007/BF00967584, Sib. Math. J. 17 (1976), 367–369. MR0412318DOI10.1007/BF00967584

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