Automorphic loops and metabelian groups

Mark Greer; Lee Raney

Commentationes Mathematicae Universitatis Carolinae (2020)

  • Volume: 61, Issue: 4, page 523-534
  • ISSN: 0010-2628

Abstract

top
Given a uniquely 2-divisible group G , we study a commutative loop ( G , ∘ ) which arises as a result of a construction in “Engelsche elemente noetherscher gruppen” (1957) by R. Baer. We investigate some general properties and applications of “ ∘ ” and determine a necessary and sufficient condition on G in order for ( G , ∘ ) to be Moufang. In “A class of loops categorically isomorphic to Bruck loops of odd order” (2014) by M. Greer, it is conjectured that G is metabelian if and only if ( G , ∘ ) is an automorphic loop. We answer a portion of this conjecture in the affirmative: in particular, we show that if G is a split metabelian group of odd order, then ( G , ∘ ) is automorphic.

How to cite

top

Greer, Mark, and Raney, Lee. "Automorphic loops and metabelian groups." Commentationes Mathematicae Universitatis Carolinae 61.4 (2020): 523-534. <http://eudml.org/doc/297022>.

@article{Greer2020,
abstract = {Given a uniquely 2-divisible group $G$, we study a commutative loop $(G,\circ )$ which arises as a result of a construction in “Engelsche elemente noetherscher gruppen” (1957) by R. Baer. We investigate some general properties and applications of “$\circ $” and determine a necessary and sufficient condition on $G$ in order for $(G, \circ )$ to be Moufang. In “A class of loops categorically isomorphic to Bruck loops of odd order” (2014) by M. Greer, it is conjectured that $G$ is metabelian if and only if $(G, \circ )$ is an automorphic loop. We answer a portion of this conjecture in the affirmative: in particular, we show that if $G$ is a split metabelian group of odd order, then $(G, \circ )$ is automorphic.},
author = {Greer, Mark, Raney, Lee},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {metabelian groups; automorphic loops; Bruck loops; Moufang loops},
language = {eng},
number = {4},
pages = {523-534},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Automorphic loops and metabelian groups},
url = {http://eudml.org/doc/297022},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Greer, Mark
AU - Raney, Lee
TI - Automorphic loops and metabelian groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 4
SP - 523
EP - 534
AB - Given a uniquely 2-divisible group $G$, we study a commutative loop $(G,\circ )$ which arises as a result of a construction in “Engelsche elemente noetherscher gruppen” (1957) by R. Baer. We investigate some general properties and applications of “$\circ $” and determine a necessary and sufficient condition on $G$ in order for $(G, \circ )$ to be Moufang. In “A class of loops categorically isomorphic to Bruck loops of odd order” (2014) by M. Greer, it is conjectured that $G$ is metabelian if and only if $(G, \circ )$ is an automorphic loop. We answer a portion of this conjecture in the affirmative: in particular, we show that if $G$ is a split metabelian group of odd order, then $(G, \circ )$ is automorphic.
LA - eng
KW - metabelian groups; automorphic loops; Bruck loops; Moufang loops
UR - http://eudml.org/doc/297022
ER -

References

top
  1. Baer R., 10.1007/BF02547953, Math. Ann. 133 (1957), 256–270 (German). MR0086815DOI10.1007/BF02547953
  2. Bender H., 10.1007/BF01899467, Arch. Math. (Basel) 18 (1967), 15–16 (German). MR0213439DOI10.1007/BF01899467
  3. Bruck R. H., A Survey of Binary Systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Heft 20, Reihe: Gruppentheorie, Springer, Berlin, 1958. Zbl0141.01401MR0093552
  4. HASH(0x23f7c08), GAP - Groups, Algorithms, Programming - a System for Computational Discrete Algebra, https://www.gap-system.org, 2008. 
  5. Glauberman G., 10.1016/0021-8693(64)90017-1, J. Algebra 1 (1964), 374–396. Zbl0155.03901MR0175991DOI10.1016/0021-8693(64)90017-1
  6. Greer M., A class of loops categorically isomorphic to Bruck loops of odd order, Comm. Algebra 42 (2014), no. 8, 3682–3697. MR3196069
  7. Isaacs I. M., 10.1090/gsm/092, Graduate Studies in Mathematics, 92, American Mathematical Society, Providence, 2008. MR2426855DOI10.1090/gsm/092
  8. Jedlička P., Kinyon M., Vojtěchovský P., 10.1090/S0002-9947-2010-05088-3, Trans. Amer. Math. Soc. 363 (2011), no. 1, 365–384. Zbl1215.20060MR2719686DOI10.1090/S0002-9947-2010-05088-3
  9. Kinyon M. K., Nagy G. P., Vojtěchovský P., 10.1016/j.jalgebra.2016.11.023, J. Algebra 473 (2017), 481–512. MR3591160DOI10.1016/j.jalgebra.2016.11.023
  10. Kinyon M. K., Vojtěchovský P., 10.1080/00927870802278917, Comm. Algebra 37 (2009), no. 4, 1428–1444. MR2510992DOI10.1080/00927870802278917
  11. McCune W. W., Prover9, Mace4, https://www.cs.unm.edu/̴ mccune/prover9/ , 2009. 
  12. Nagy G. P., Vojtěchovský P., Computing with small quasigroups and loops, Quasigroups Related Systems 15 (2007), no. 1, 77–94. MR2379126
  13. Stuhl I., Vojtěchovský P., 10.1090/conm/721/14510, Nonassociative Mathematics and Its Applications, Contemp. Math., 721, Amer. Math. Soc., Providence, 2019, pages 261–276. MR3898514DOI10.1090/conm/721/14510
  14. Pflugfelder H. O., Quasigroups and Loops: Introduction, Sigma Series in Pure Mathematics, 7, Heldermann Verlag, Berlin, 1990. Zbl0715.20043MR1125767
  15. Thompson J. G., 10.1007/BF01111272, Math. Z. 86 (1964), 12–13. MR0168653DOI10.1007/BF01111272

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.