Moufang semidirect products of loops with groups and inverse property extensions

Mark Greer; Lee Raney

Commentationes Mathematicae Universitatis Carolinae (2014)

  • Volume: 55, Issue: 3, page 411-420
  • ISSN: 0010-2628

Abstract

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We investigate loops which can be written as the semidirect product of a loop and a group, and we provide a necessary and sufficient condition for such a loop to be Moufang. We also examine a class of loop extensions which arise as a result of a finite cyclic group acting as a group of semiautomorphisms on an inverse property loop. In particular, we consider closure properties of certain extensions similar to those as in [S. Gagola III, Cyclic extensions of Moufang loops induced by semiautomorphisms, J. Algebra Appl. 13 (2014), no. 4, 1350128], but from an external point of view.

How to cite

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Greer, Mark, and Raney, Lee. "Moufang semidirect products of loops with groups and inverse property extensions." Commentationes Mathematicae Universitatis Carolinae 55.3 (2014): 411-420. <http://eudml.org/doc/261866>.

@article{Greer2014,
abstract = {We investigate loops which can be written as the semidirect product of a loop and a group, and we provide a necessary and sufficient condition for such a loop to be Moufang. We also examine a class of loop extensions which arise as a result of a finite cyclic group acting as a group of semiautomorphisms on an inverse property loop. In particular, we consider closure properties of certain extensions similar to those as in [S. Gagola III, Cyclic extensions of Moufang loops induced by semiautomorphisms, J. Algebra Appl. 13 (2014), no. 4, 1350128], but from an external point of view.},
author = {Greer, Mark, Raney, Lee},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {extensions; semidirect products; Moufang loops; inverse property loops; Moufang loops; semidirect products; inverse property loops; loop extensions},
language = {eng},
number = {3},
pages = {411-420},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Moufang semidirect products of loops with groups and inverse property extensions},
url = {http://eudml.org/doc/261866},
volume = {55},
year = {2014},
}

TY - JOUR
AU - Greer, Mark
AU - Raney, Lee
TI - Moufang semidirect products of loops with groups and inverse property extensions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2014
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 55
IS - 3
SP - 411
EP - 420
AB - We investigate loops which can be written as the semidirect product of a loop and a group, and we provide a necessary and sufficient condition for such a loop to be Moufang. We also examine a class of loop extensions which arise as a result of a finite cyclic group acting as a group of semiautomorphisms on an inverse property loop. In particular, we consider closure properties of certain extensions similar to those as in [S. Gagola III, Cyclic extensions of Moufang loops induced by semiautomorphisms, J. Algebra Appl. 13 (2014), no. 4, 1350128], but from an external point of view.
LA - eng
KW - extensions; semidirect products; Moufang loops; inverse property loops; Moufang loops; semidirect products; inverse property loops; loop extensions
UR - http://eudml.org/doc/261866
ER -

References

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  1. Bruck R.H., 10.1090/S0002-9939-1952-0047635-6, Proc. Amer. Math. Soc. 2 (1952), 66–72. Zbl0046.01803MR0047635DOI10.1090/S0002-9939-1952-0047635-6
  2. Bruck R.H., A Survey of Binary Systems, Springer, Berlin, 1971. Zbl0141.01401MR0093552
  3. Chein O., 10.1090/S0002-9947-1974-0330336-3, Trans. Amer. Math. Soc. 188 (1974), 31–51. Zbl0286.20088MR0330336DOI10.1090/S0002-9947-1974-0330336-3
  4. Chein O., Moufang loops of small order, Mem. Amer. Math. Soc. 13 (1978), no. 197. Zbl0378.20053MR0466391
  5. Greer M., Semiautomorphic inverse property loops, submitted. 
  6. The GAP Group, Groups, Algorithms, and Programming, http://www.gap-system.org (2008). 
  7. Gagola S. III, 10.1142/S0219498813501284, J. Algebra Appl. 13 (2014), no. 4, 1350128. MR3153863DOI10.1142/S0219498813501284
  8. Kinyon M.K., Jones O., 10.1080/00927870008827079, Comm. Algebra 28 (2000), 4137–4164. Zbl0974.20049MR1772003DOI10.1080/00927870008827079
  9. Kinyon M.K., Kunen K., Phillips J.D., 10.1007/s00012-002-8205-0, Algebra Universalis 48 (2002), 81–101. Zbl1058.20057MR1930034DOI10.1007/s00012-002-8205-0
  10. McCune W.W., Prover9, Mace4, http://www.cs.unm.edu/ mccune/prover9/ (2009). 
  11. Nagy G.P., Vojtěchovský P., Loops: Computing with quasigroups and loops, http://www.math.du.edu/loops (2008). 
  12. Pflugfelder H.O., Quasigroups and Loops: Introduction, Sigma Series in Pure Mathematics, 7, Heldermann, Berlin, 1990. Zbl0715.20043MR1125767

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