| Title:
|
On central nilpotency in finite loops with nilpotent inner mapping groups (English) |
| Author:
|
Niemenmaa, Markku |
| Author:
|
Rytty, Miikka |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
49 |
| Issue:
|
2 |
| Year:
|
2008 |
| Pages:
|
271-277 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we consider finite loops whose inner mapping groups are nilpotent. We first consider the case where the inner mapping group $I(Q)$ of a loop $Q$ is the direct product of a dihedral group of order $8$ and an abelian group. Our second result deals with the case where $Q$ is a $2$-loop and $I(Q)$ is a nilpotent group whose nonabelian Sylow subgroups satisfy a special condition. In both cases it turns out that $Q$ is centrally nilpotent. (English) |
| Keyword:
|
loop |
| Keyword:
|
group |
| Keyword:
|
connected transversals |
| MSC:
|
20D10 |
| MSC:
|
20D15 |
| MSC:
|
20D35 |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 1192.20057 |
| idMR:
|
MR2426891 |
| . |
| Date available:
|
2009-05-05T17:11:13Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119721 |
| . |
| Reference:
|
[1] Amberg B., Franciosi S., de Giovanni F.: Products of Groups.Clarendon Press, New York, 1992. Zbl 0774.20001, MR 1211633 |
| Reference:
|
[2] Bruck R.: Contributions to the theory of loops.Trans. Amer. Math. Soc. 60 (1946), 245-354. Zbl 0061.02201, MR 0017288 |
| Reference:
|
[3] Huppert B.: Endliche Gruppen I.Springer, Berlin-New York, 1967. Zbl 0412.20002, MR 0224703 |
| Reference:
|
[4] Kepka T.: On the abelian inner permutations groups of loops.Comm. Algebra 26 (1998), 857-861. MR 1606178, 10.1080/00927879808826169 |
| Reference:
|
[5] Kepka T., Niemenmaa M.: On loops with cyclic inner mapping groups.Arch. Math. 60 (1993), 233-236. MR 1201636, 10.1007/BF01198806 |
| Reference:
|
[6] Niemenmaa M.: On loops which have dihedral $2$-groups as inner mapping groups.Bull. Austral. Math. Soc. 52 (1995), 153-160. Zbl 0838.20080, MR 1344268, 10.1017/S0004972700014520 |
| Reference:
|
[7] Niemenmaa M.: On finite loops and their inner mapping groups.Comment. Math. Univ. Carolin. 45 (2004), 341-347. Zbl 1101.20045, MR 2075281 |
| Reference:
|
[8] 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:
|
[9] 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 |
| Reference:
|
[10] Niemenmaa M., Kepka T.: On connected transversals to abelian subgroups.Bull. Austral. Math. Soc. 49 (1994), 121-128. Zbl 0799.20020, MR 1262682, 10.1017/S0004972700016166 |
| . |
