| Title:
|
Locally soluble-by-finite groups with small deviation for non-subnormal subgroups (English) |
| Author:
|
Kurdachenko, Leonid A. |
| Author:
|
Smith, Howard |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
48 |
| Issue:
|
1 |
| Year:
|
2007 |
| Pages:
|
1-7 |
| . |
| Category:
|
math |
| . |
| Summary:
|
A group $G$ has subnormal deviation at most $1$ if, for every descending chain $H_{0}>H_{1}>\dots $ of non-subnormal subgroups of $G$, for all but finitely many $i$ there is no infinite descending chain of non-subnormal subgroups of $G$ that contain $H_{i+1}$ and are contained in $H_{i}$. This property $\frak P$, say, was investigated in a previous paper by the authors, where soluble groups with $\frak P$ and locally nilpotent groups with $\frak P$ were effectively classified. The present article affirms a conjecture from that article by showing that locally soluble-by-finite groups with $\frak P$ are soluble-by-finite and are therefore classified. (English) |
| Keyword:
|
subnormal subgroups |
| Keyword:
|
locally soluble-by-finite groups |
| MSC:
|
20E15 |
| MSC:
|
20F14 |
| MSC:
|
20F19 |
| idZBL:
|
Zbl 1174.20309 |
| idMR:
|
MR2338825 |
| . |
| Date available:
|
2009-05-05T17:00:57Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119634 |
| . |
| Reference:
|
[1] Dixon M.R., Evans M.J., Smith H.: Locally (soluble-by-finite) groups with various restrictions on subgroups of infinite rank.Glasgow J. Math. 47 (2005), 309-317. Zbl 1088.20013, MR 2203498 |
| Reference:
|
[2] Kegel O.H., Wehfritz B.A.F.: Locally Finite Groups.North-Holland, Amsterdam-London, 1973. |
| Reference:
|
[3] Kropholler P.: On finitely generated soluble groups with no large wreath product sections.Proc. London Math. Soc. 49 (1984), 155-169. Zbl 0537.20013, MR 0743376 |
| Reference:
|
[4] Kurdachenko L.A., Smith H.: Groups with the weak minimal condition for non-subnormal subgroups.Ann. Mat. Pura Appl. (4) 173 (1997), 299-312. Zbl 0939.20040, MR 1625608 |
| Reference:
|
[5] Kurdachenko L.A., Smith H.: Groups with the weak minimal condition for non-subnormal subgroups II.Comment. Math. Univ. Carolin. 46 (2005), 601-605. Zbl 1106.20023, MR 2259493 |
| Reference:
|
[6] Kurdachenko L.A., Smith H.: Groups with small deviation for non-subnormal subgroups.preprint. MR 2506959 |
| Reference:
|
[7] Möhres W.: Auflösbarkeit von Gruppen, deren Untergruppen alle subnormal sind.Arch. Math. (Basel) 54 (1990), 232-235. MR 1037610 |
| Reference:
|
[8] Robinson D.J.S.: Finiteness Conditions and Generalized Soluble Groups.2 vols., Springer, New York-Berlin, 1972. Zbl 0243.20033 |
| . |
