| Title:
|
Moufang loops of odd order $p_1p_2\dots p_nq^3$ with non-trivial nucleus (English) |
| Author:
|
Rajah, Andrew |
| Author:
|
Chong, Kam-Yoon |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
49 |
| Issue:
|
2 |
| Year:
|
2008 |
| Pages:
|
301-307 |
| . |
| Category:
|
math |
| . |
| Summary:
|
It has been proven by F. Leong and the first author (J. Algebra {\bf 190} (1997), 474--486) that all Moufang loops of order $p^\alpha q_1^{\beta_1}q_2^{\beta_2}\cdot \cdot \cdot q_n^{\beta_n}$ where $p$ and $q_i$ are odd primes, are associative if $p<q_1<q_2<\cdot \cdot \cdot<q_n$, and \roster \item"(i)" $\alpha\leq 3$, $\beta_i\leq 2$; or \item"(ii)" $p\geq 5$, $\alpha\leq 4$, $\beta_i\leq2$. \endroster The first author also proved that if $p$ and $q$ are distinct odd primes, then all Moufang loops of order $pq^3$ are associative if and only if $q\not\equiv 1(\text{\rm mod}\, p)$ (J. Algebra {\bf 235} (2001), 66--93). In this paper, we prove that all Moufang loops of order $p_1p_2\cdot \cdot \cdot p_nq^3$ where $p_i$ and $q$ are odd primes, are associative if $p_1<p_2<\cdot \cdot \cdot <p_n<q$, $q\not\equiv 1(\text{\rm mod}\, p_i)$, $p_i\not\equiv 1(\text{\rm mod}\, p_j)$ and the nucleus is not trivial. (English) |
| Keyword:
|
Moufang loop |
| Keyword:
|
order |
| Keyword:
|
nonassociative |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 1192.20061 |
| idMR:
|
MR2426894 |
| . |
| Date available:
|
2009-05-05T17:11:28Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119724 |
| . |
| Reference:
|
[1] Bruck R.H.: A Survey of Binary Systems.Springer, New York, 1971. Zbl 0141.01401, MR 0093552 |
| Reference:
|
[2] Chein O.: Moufang loops of small order I.Trans. Amer. Math. Soc. 188 2 (1974), 31-51. Zbl 0286.20088, MR 0330336, 10.1090/S0002-9947-1974-0330336-3 |
| Reference:
|
[3] Chein O.: Moufang loops of small order.Memoirs Amer. Math. Soc. 13 197 (1978), 1-131. Zbl 0378.20053, MR 0466391 |
| Reference:
|
[4] Chein O., Rajah A.: Possible orders of nonassociative Moufang loops.Comment. Math. Univ. Carolin. 41 2 (2000), 237-244. Zbl 1038.20045, MR 1780867 |
| Reference:
|
[5] Glauberman G.: On loops of odd order II.J. Algebra 8 (1968), 393-414. Zbl 0155.03901, MR 0222198, 10.1016/0021-8693(68)90050-1 |
| Reference:
|
[6] Grishkov A.N., Zavarnitsine A.V.: Lagrange's Theorem for Moufang loops.Math. Proc. Cambridge Philos. Soc. 139 (2005), 41-57. Zbl 1091.20039, MR 2155504, 10.1017/S0305004105008388 |
| Reference:
|
[7] Herstein I.N.: Topics in Algebra.John Wiley & Sons, Inc., New York, 1975. Zbl 0122.01301, MR 0171801 |
| Reference:
|
[8] Leong F., Rajah A.: On Moufang loops of odd order $pq^2$.J. Algebra 176 (1995), 265-270. MR 1345304, 10.1006/jabr.1995.1243 |
| Reference:
|
[9] Leong F., Rajah A.: Moufang loops of odd order $p_1^2p_2^2\cdots p_m^2$.J. Algebra 181 (1996), 876-883. MR 1386583, 10.1006/jabr.1996.0150 |
| Reference:
|
[10] Leong F., Rajah A.: Moufang loops of odd order $p^4q_1\cdots q_n$.J. Algebra 184 (1996), 561-569. Zbl 0860.20054, MR 1409228, 10.1006/jabr.1996.0274 |
| Reference:
|
[11] Leong F., Rajah A.: Moufang loops of odd order $p^\alpha q_1^2\cdots q_n^2r_1\cdots r_m$.J. Algebra 190 (1997), 474-486. Zbl 0874.20046, MR 1441958 |
| Reference:
|
[12] Leong F., Rajah A.: Split extension in Moufang loops.Publ. Math. Debrecen 52 1-2 (1998), 33-42. MR 1603303 |
| Reference:
|
[13] Purtill M.: On Moufang loops of order the product of three odd primes.J. Algebra 112 (1988), 122-128. Zbl 0644.20040, MR 0921968, 10.1016/0021-8693(88)90136-6 |
| Reference:
|
[14] Purtill M.: Corrigendum.J. Algebra 145 (1992), 262. Zbl 0742.20068, MR 1144674, 10.1016/0021-8693(92)90192-O |
| Reference:
|
[15] Rajah A.: Moufang loops of odd order $pq^3$.J. Algebra 235 (2001), 66-93. Zbl 0973.20062, MR 1807655, 10.1006/jabr.2000.8422 |
| . |
