| Title:
|
On the uniqueness of loops $M(G,2)$ (English) |
| Author:
|
Vojtěchovský, Petr |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
44 |
| Issue:
|
4 |
| Year:
|
2003 |
| Pages:
|
629-635 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $G$ be a finite group and $C_2$ the cyclic group of order 2. Consider the 8 multiplicative operations $(x,y)\mapsto (x^iy^j)^k$, where $i,j,k\in\{-1,\,1\}$. Define a new multiplication on $G\times C_2$ by assigning one of the above 8 multiplications to each quarter $(G\times\{i\})\times(G\times\{j\})$, for $i,j\in C_2$. If the resulting quasigroup is a Bol loop, it is Moufang. When $G$ is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomorphic, known as loops $M(G,2)$. (English) |
| Keyword:
|
Moufang loops |
| Keyword:
|
loops $M(G, 2)$ |
| Keyword:
|
inverse property loops |
| Keyword:
|
Bol loops |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 1101.20047 |
| idMR:
|
MR2062879 |
| . |
| Date available:
|
2009-01-08T19:31:43Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119417 |
| . |
| Reference:
|
[1] Chein O.: Moufang loops of small order.Memoirs of the American Mathematical Society, Volume 13, Issue 1, Number 197 (1978). Zbl 0378.20053, MR 0466391 |
| Reference:
|
[2] Chein O., Pflugfelder H.O., Smith J.D.H.: Quasigroups and Loops: Theory and Applications.Sigma Series in Pure Mathematics 8, Heldermann Verlag, Berlin, 1990. Zbl 0719.20036, MR 1125806 |
| Reference:
|
[3] Chein O., Pflugfelder H.O.: The smallest Moufang loop.Arch. Math. 22 (1971), 573-576. Zbl 0241.20061, MR 0297914 |
| Reference:
|
[4] Drápal A., Vojtěchovský P.: Moufang loops that share associator and three quarters of their multiplication tables.submitted. |
| Reference:
|
[5] Goodaire E.G., May S., Raman M.: The Moufang Loops of Order less than $64$.Nova Science Publishers, 1999. Zbl 0964.20043, MR 1689624 |
| Reference:
|
[6] Pflugfelder H.O.: Quasigroups and Loops: Introduction.Sigma Series in Pure Mathematics 7, Heldermann Verlag, Berlin, 1990. Zbl 0715.20043, MR 1125767 |
| Reference:
|
[7] Vojtěchovský P.: The smallest Moufang loop revisited.to appear in Results Math. MR 2011917 |
| Reference:
|
[8] Vojtěchovský P.: Connections between codes, groups and loops.Ph.D. Thesis, Charles Univesity, 2003. |
| . |
