Title:
|
Schreier loops (English) |
Author:
|
Nagy, Péter T. |
Author:
|
Strambach, Karl |
Language:
|
English |
Journal:
|
Czechoslovak Mathematical Journal |
ISSN:
|
0011-4642 (print) |
ISSN:
|
1572-9141 (online) |
Volume:
|
58 |
Issue:
|
3 |
Year:
|
2008 |
Pages:
|
759-786 |
Summary lang:
|
English |
. |
Category:
|
math |
. |
Summary:
|
We study systematically the natural generalization of Schreier's extension theory to obtain proper loops and show that this construction gives a rich family of examples of loops in all traditional common, important loop classes. (English) |
Keyword:
|
extension of loops |
Keyword:
|
non-associative extension of groups |
Keyword:
|
weak associativity properties of extensions |
Keyword:
|
central extensions |
MSC:
|
20N05 |
idZBL:
|
Zbl 1166.20058 |
idMR:
|
MR2455937 |
. |
Date available:
|
2010-07-20T14:06:56Z |
Last updated:
|
2020-07-03 |
Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140420 |
. |
Reference:
|
[1] Birkenmeier, G., Davis, B., Reeves, K., Xiao, S.: Is a semidirect product of groups necessarily a group?.Proc. Amer. Math. Soc. 118 (1993), 689-692. Zbl 0782.20053, MR 1157998, 10.1090/S0002-9939-1993-1157998-2 |
Reference:
|
[2] Birkenmeier, G., Xiao, S.: Loops which are semidirect products of groups.Comm. in Algebra 23 (1995), 81-95. Zbl 0821.20059, MR 1311775, 10.1080/00927879508825207 |
Reference:
|
[3] Chein, O.: Moufang loops of small order I.Trans. Amer. Math. Soc. 188 (1974), 31-51. Zbl 0286.20088, MR 0330336, 10.1090/S0002-9947-1974-0330336-3 |
Reference:
|
[4] Chein, O.: Moufang loops of small order.Mem. Amer. Math. Soc. 13 (1978), 31-51. Zbl 0378.20053, MR 0466391 |
Reference:
|
[5] Chein, O.: Examples and methods of construction, Chapter II in Quasigroups and Loops: Theory and Applications.O. Chein, H. O. Pflugfelder, J. D. H. Smith 27-93 Heldermann Verlag, Berlin (1990). MR 1125808 |
Reference:
|
[6] Chein, O., Goodaire, E. G.: A new construction of Bol loops of order $8k$.Journal of Algebra 287 (2005), 103-122. Zbl 1084.20042, MR 2134260, 10.1016/j.jalgebra.2005.01.031 |
Reference:
|
[7] Csörgő, P., Drápal, A.: Left conjugacy closed loops of nilpotency class two.Results Math. 47 (2005), 242-265. MR 2153496, 10.1007/BF03323028 |
Reference:
|
[8] Figula, A.: $3$-dimensional Bol loops as sections in non-solvable Lie groups.Forum Math. 17 (2005), 431-460. Zbl 1085.53041, MR 2138500, 10.1515/form.2005.17.3.431 |
Reference:
|
[9] Figula, A.: $3$-dimensional loops on non-solvable reductive spaces.Adv. Geom. 5 (2005), 391-420. Zbl 1087.53049, MR 2154833, 10.1515/advg.2005.5.3.391 |
Reference:
|
[10] Figula, A.: $3$-dimensional Bol loops corresponding to solvable Lie triple systems.Publ. Math. Debrecen 70 (2007), 59-101. Zbl 1174.22001, MR 2288468 |
Reference:
|
[11] Figula, A., Strambach, K.: Loops which are semidirect products of groups.Acta Math. Hungar. 114 (2007), 247-266. Zbl 1123.20061, MR 2296546, 10.1007/s10474-006-0529-3 |
Reference:
|
[12] Kinyon, M. K., Kunen, K.: The structure of extra loops.Quasigroups Related Systems 12 (2004), 39-60. Zbl 1076.20065, MR 2130578 |
Reference:
|
[13] Kinyon, M. K., Kunen, K., Phillips, J. D.: A generalization of Moufang and Steiner loops.Algebra Universalis 48 (2002), 81-101. Zbl 1058.20057, MR 1930034, 10.1007/s00012-002-8205-0 |
Reference:
|
[14] Kinyon, M. K., Phillips, J. D., Vojtěchovský, P.: C-loops; extensions and constructions.J. Algebra Appl. 6 (2007), 1-20. MR 2302693, 10.1142/S0219498807001990 |
Reference:
|
[15] Kinyon, M. K., Phillips, J. D., Vojtěchovský, P.: When is the commutant of a Bol loop a subloop.Trans. Amer. Math. Soc. 360 (2008), 2393-2408. MR 2373318, 10.1090/S0002-9947-07-04391-7 |
Reference:
|
[16] Kurosh, A. G.: The Theory of Groups, Vol. 2, transl. from Russian by K. A. Hirsch.Chelsea Publishing Co., New York (1956). MR 0109842 |
Reference:
|
[17] Nagy, G. P.: Algebraic commutative Moufang loops.Forum Math. 15 (2003), 37-62. Zbl 1084.20515, MR 1957278 |
Reference:
|
[18] Nagy, P. T., Strambach, K.: Loops in Group Theory and Lie Theory, Expositions in Mathematics 35.Walter de Gruyter, Berlin-New York (2002). MR 1899331 |
Reference:
|
[19] Pflugfelder, H. O.: Quasigroups and Loops: An Introduction.Heldermann Verlag, Berlin (1990). MR 1125767 |
Reference:
|
[20] Schreier, O.: Über die Erweiterung von Gruppen I.Monatshefte f. Math. 34 (1926), 165-180. MR 1549403 |
Reference:
|
[21] Schreier, O.: Über die Erweiterung von Gruppen II.Abh. Math. Sem. Univ. Hamburg 4 (1926), 321-346. 10.1007/BF02950735 |
Reference:
|
[22] Suvorov, N. M., Krjuckov, N. I.: Examples of certain quasigroups and loops that permit only the discrete topologization.Russian Sib. Mat. Zh. 17 (1976), 471-473. MR 0412318 |
Reference:
|
[23] Suvorov, N. M.: A commutative IP-loop that admits only discrete topologization.Russian Sib. Mat. Zh. 32 (1991), 193. MR 1155817 |
Reference:
|
[24] Suzuki, M.: Group Theory, Vol. 1.Grundlehren der Mathematischen Wissenschaften 10, Springer Verlag, Berlin-New York (1982). MR 0648772 |
Reference:
|
[25] Winterroth, E.: Right Bol loops with a finite dimensional group of multiplications.Publ. Math. Debrecen 59 (2001), 161-173. Zbl 1012.22005, MR 1853499 |
. |