Previous |  Up |  Next

Article

Keywords:
extension of loops; non-associative extension of groups; weak associativity properties of extensions; central extensions
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.
References:
[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. DOI 10.1090/S0002-9939-1993-1157998-2 | MR 1157998 | Zbl 0782.20053
[2] Birkenmeier, G., Xiao, S.: Loops which are semidirect products of groups. Comm. in Algebra 23 (1995), 81-95. DOI 10.1080/00927879508825207 | MR 1311775 | Zbl 0821.20059
[3] Chein, O.: Moufang loops of small order I. Trans. Amer. Math. Soc. 188 (1974), 31-51. DOI 10.1090/S0002-9947-1974-0330336-3 | MR 0330336 | Zbl 0286.20088
[4] Chein, O.: Moufang loops of small order. Mem. Amer. Math. Soc. 13 (1978), 31-51. MR 0466391 | Zbl 0378.20053
[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
[6] Chein, O., Goodaire, E. G.: A new construction of Bol loops of order $8k$. Journal of Algebra 287 (2005), 103-122. DOI 10.1016/j.jalgebra.2005.01.031 | MR 2134260 | Zbl 1084.20042
[7] Csörgő, P., Drápal, A.: Left conjugacy closed loops of nilpotency class two. Results Math. 47 (2005), 242-265. DOI 10.1007/BF03323028 | MR 2153496
[8] Figula, A.: $3$-dimensional Bol loops as sections in non-solvable Lie groups. Forum Math. 17 (2005), 431-460. DOI 10.1515/form.2005.17.3.431 | MR 2138500 | Zbl 1085.53041
[9] Figula, A.: $3$-dimensional loops on non-solvable reductive spaces. Adv. Geom. 5 (2005), 391-420. DOI 10.1515/advg.2005.5.3.391 | MR 2154833 | Zbl 1087.53049
[10] Figula, A.: $3$-dimensional Bol loops corresponding to solvable Lie triple systems. Publ. Math. Debrecen 70 (2007), 59-101. MR 2288468 | Zbl 1174.22001
[11] Figula, A., Strambach, K.: Loops which are semidirect products of groups. Acta Math. Hungar. 114 (2007), 247-266. DOI 10.1007/s10474-006-0529-3 | MR 2296546 | Zbl 1123.20061
[12] Kinyon, M. K., Kunen, K.: The structure of extra loops. Quasigroups Related Systems 12 (2004), 39-60. MR 2130578 | Zbl 1076.20065
[13] Kinyon, M. K., Kunen, K., Phillips, J. D.: A generalization of Moufang and Steiner loops. Algebra Universalis 48 (2002), 81-101. DOI 10.1007/s00012-002-8205-0 | MR 1930034 | Zbl 1058.20057
[14] Kinyon, M. K., Phillips, J. D., Vojtěchovský, P.: C-loops; extensions and constructions. J. Algebra Appl. 6 (2007), 1-20. DOI 10.1142/S0219498807001990 | MR 2302693
[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. DOI 10.1090/S0002-9947-07-04391-7 | MR 2373318
[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
[17] Nagy, G. P.: Algebraic commutative Moufang loops. Forum Math. 15 (2003), 37-62. MR 1957278 | Zbl 1084.20515
[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
[19] Pflugfelder, H. O.: Quasigroups and Loops: An Introduction. Heldermann Verlag, Berlin (1990). MR 1125767
[20] Schreier, O.: Über die Erweiterung von Gruppen I. Monatshefte f. Math. 34 (1926), 165-180. MR 1549403
[21] Schreier, O.: Über die Erweiterung von Gruppen II. Abh. Math. Sem. Univ. Hamburg 4 (1926), 321-346. DOI 10.1007/BF02950735
[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
[23] Suvorov, N. M.: A commutative IP-loop that admits only discrete topologization. Russian Sib. Mat. Zh. 32 (1991), 193. MR 1155817
[24] Suzuki, M.: Group Theory, Vol. 1. Grundlehren der Mathematischen Wissenschaften 10, Springer Verlag, Berlin-New York (1982). MR 0648772
[25] Winterroth, E.: Right Bol loops with a finite dimensional group of multiplications. Publ. Math. Debrecen 59 (2001), 161-173. MR 1853499 | Zbl 1012.22005
Partner of
EuDML logo