| Title:
|
On loops that are abelian groups over the nucleus and Buchsteiner loops (English) |
| Author:
|
Csörgö, Piroska |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
49 |
| Issue:
|
2 |
| Year:
|
2008 |
| Pages:
|
197-208 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
conjugacy closed loops |
| Keyword:
|
Buchsteiner loops |
| MSC:
|
20D10 |
| MSC:
|
20D15 |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 1192.20052 |
| idMR:
|
MR2426885 |
| . |
| Date available:
|
2009-05-05T17:07:34Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119715 |
| . |
| Reference:
|
[1] Bruck R.H.: Contributions to the theory of loops.Trans. Amer. Math. Soc. 60 (1946), 245-354. Zbl 0061.02201, MR 0017288 |
| Reference:
|
[2] Buchsteiner H.H.: O nekotorom klasse binarnych lup.Mat. Issled. 39 (1976), 54-66. |
| Reference:
|
[3] Csörgö P., Drápal A.: On loops rich in automorphisms that are abelian modulo the nucleus.submitted. |
| Reference:
|
[4] Csörgö P.: On connected transversals to abelian subgroups and loop theoretical consequences.Arch. Math. (Basel) 86 (2006), 499-516. Zbl 1113.20055, MR 2241599, 10.1007/s00013-006-1379-5 |
| Reference:
|
[5] Csörgö P., Drápal A.: Left conjugacy closed loops of nilpotency class two.Results Math. 47 (2005), 242-265. Zbl 1097.20053, MR 2153496, 10.1007/BF03323028 |
| Reference:
|
[6] Csörgö P., Drápal A., Kinyon M.K.: Buchsteiner loops.submitted. |
| Reference:
|
[7] Csörgö P., Drápal A.: Buchsteiner loops and conjugacy closedness.submitted. |
| Reference:
|
[8] Csörgö P.: Abelian inner mappings and nilpotency class greater than two.European J. Combin. 28 (2007), 858-867. Zbl 1149.20053, MR 2300766, 10.1016/j.ejc.2005.12.002 |
| Reference:
|
[9] Drápal A., Kinyon M.: Buchsteiner loops: Associators and constructions.submitted. |
| Reference:
|
[10] Drápal A., Jedlička P.: On loop identities that can be obtained by a nuclear identification.submitted. |
| Reference:
|
[11] Drápal A., Vojtěchovský P.: Explicit constructions of loops with commuting inner mappings.submitted. |
| Reference:
|
[12] Nagy G.P., Vojtěchovský P.: Moufang groups with commuting inner mappings.submitted. |
| Reference:
|
[13] Niemenmaa M., Kepka T.: On multiplication groups of loops.J. Algebra 135 (1990), 112-122. Zbl 0706.20046, MR 1076080, 10.1016/0021-8693(90)90152-E |
| Reference:
|
[14] Niemenmaa M., Kepka T.: On connected transversals to abelian subgroups in finite groups.Bull. London Math. Soc. 24 (1992), 343-346. Zbl 0793.20064, MR 1165376, 10.1112/blms/24.4.343 |
| . |
