On loops that are abelian groups over the nucleus and Buchsteiner loops

Piroska Csörgö

Commentationes Mathematicae Universitatis Carolinae (2008)

  • Volume: 49, Issue: 2, page 197-208
  • ISSN: 0010-2628

Abstract

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We give sufficient and in some cases necessary conditions for the conjugacy closedness of Q / Z ( Q ) provided the commutativity of Q / N . We show that if for some loop Q , Q / N and Inn Q are abelian groups, then Q / Z ( Q ) is a CC loop, consequently Q has nilpotency class at most three. We give additionally some reasonable conditions which imply the nilpotency of the multiplication group of class at most three. We describe the structure of Buchsteiner loops with abelian inner mapping groups.

How to cite

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Csörgö, Piroska. "On loops that are abelian groups over the nucleus and Buchsteiner loops." Commentationes Mathematicae Universitatis Carolinae 49.2 (2008): 197-208. <http://eudml.org/doc/250458>.

@article{Csörgö2008,
abstract = {We give sufficient and in some cases necessary conditions for the conjugacy closedness of $Q/Z(Q)$ provided the commutativity of $Q/N$. We show that if for some loop $Q$, $Q/N$ and $\operatorname\{Inn\} Q$ are abelian groups, then $Q/Z(Q)$ is a CC loop, consequently $Q$ has nilpotency class at most three. We give additionally some reasonable conditions which imply the nilpotency of the multiplication group of class at most three. We describe the structure of Buchsteiner loops with abelian inner mapping groups.},
author = {Csörgö, Piroska},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {conjugacy closed loops; Buchsteiner loops; conjugacy closed loops; Buchsteiner loops; conjugacy closedness; CC-loops; nilpotency of multiplication groups; Abelian inner mapping groups},
language = {eng},
number = {2},
pages = {197-208},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On loops that are abelian groups over the nucleus and Buchsteiner loops},
url = {http://eudml.org/doc/250458},
volume = {49},
year = {2008},
}

TY - JOUR
AU - Csörgö, Piroska
TI - On loops that are abelian groups over the nucleus and Buchsteiner loops
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2008
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 49
IS - 2
SP - 197
EP - 208
AB - We give sufficient and in some cases necessary conditions for the conjugacy closedness of $Q/Z(Q)$ provided the commutativity of $Q/N$. We show that if for some loop $Q$, $Q/N$ and $\operatorname{Inn} Q$ are abelian groups, then $Q/Z(Q)$ is a CC loop, consequently $Q$ has nilpotency class at most three. We give additionally some reasonable conditions which imply the nilpotency of the multiplication group of class at most three. We describe the structure of Buchsteiner loops with abelian inner mapping groups.
LA - eng
KW - conjugacy closed loops; Buchsteiner loops; conjugacy closed loops; Buchsteiner loops; conjugacy closedness; CC-loops; nilpotency of multiplication groups; Abelian inner mapping groups
UR - http://eudml.org/doc/250458
ER -

References

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  1. Bruck R.H., Contributions to the theory of loops, Trans. Amer. Math. Soc. 60 (1946), 245-354. (1946) Zbl0061.02201MR0017288
  2. Buchsteiner H.H., O nekotorom klasse binarnych lup, Mat. Issled. 39 (1976), 54-66. (1976) 
  3. Csörgö P., Drápal A., On loops rich in automorphisms that are abelian modulo the nucleus, submitted. 
  4. Csörgö P., 10.1007/s00013-006-1379-5, Arch. Math. (Basel) 86 (2006), 499-516. (2006) Zbl1113.20055MR2241599DOI10.1007/s00013-006-1379-5
  5. Csörgö P., Drápal A., 10.1007/BF03323028, Results Math. 47 (2005), 242-265. (2005) Zbl1097.20053MR2153496DOI10.1007/BF03323028
  6. Csörgö P., Drápal A., Kinyon M.K., Buchsteiner loops, submitted. 
  7. Csörgö P., Drápal A., Buchsteiner loops and conjugacy closedness, submitted. 
  8. Csörgö P., 10.1016/j.ejc.2005.12.002, European J. Combin. 28 (2007), 858-867. (2007) Zbl1149.20053MR2300766DOI10.1016/j.ejc.2005.12.002
  9. Drápal A., Kinyon M., Buchsteiner loops: Associators and constructions, submitted. 
  10. Drápal A., Jedlička P., On loop identities that can be obtained by a nuclear identification, submitted. 
  11. Drápal A., Vojtěchovský P., Explicit constructions of loops with commuting inner mappings, submitted. 
  12. Nagy G.P., Vojtěchovský P., Moufang groups with commuting inner mappings, submitted. 
  13. Niemenmaa M., Kepka T., 10.1016/0021-8693(90)90152-E, J. Algebra 135 (1990), 112-122. (1990) Zbl0706.20046MR1076080DOI10.1016/0021-8693(90)90152-E
  14. Niemenmaa M., Kepka T., 10.1112/blms/24.4.343, Bull. London Math. Soc. 24 (1992), 343-346. (1992) Zbl0793.20064MR1165376DOI10.1112/blms/24.4.343

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