| Title:
|
Every $2$-group with all subgroups normal-by-finite is locally finite (English) |
| Author:
|
Jabara, Enrico |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
2 |
| Year:
|
2018 |
| Pages:
|
491-496 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A group $G$ has all of its subgroups normal-by-finite if $H/H_{G}$ is finite for all subgroups $H$ of $G$. The Tarski-groups provide examples of $p$-groups ($p$ a ``large'' prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a $2$-group with every subgroup normal-by-finite is locally finite. We also prove that if $| H/H_{G} | \leq 2$ for every subgroup $H$ of $G$, then $G$ contains an Abelian subgroup of index at most $8$. (English) |
| Keyword:
|
$2$-group |
| Keyword:
|
locally finite group |
| Keyword:
|
normal-by-finite subgroup |
| Keyword:
|
core-finite group |
| MSC:
|
20D15 |
| MSC:
|
20F14 |
| MSC:
|
20F50 |
| idZBL:
|
Zbl 06890385 |
| idMR:
|
MR3819186 |
| DOI:
|
10.21136/CMJ.2018.0504-16 |
| . |
| Date available:
|
2018-06-11T10:55:59Z |
| Last updated:
|
2020-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147231 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[5] Ol'shanskiĭ, A. Yu.: Geometry of Defining Relations in Groups.Mathematics and Its Applications. Soviet Series 70, Kluwer Academic Publishers, Dordrecht (1991). Zbl 0732.20019, MR 1191619, 10.1007/978-94-011-3618-1 |
| Reference:
|
[6] Robinson, D. J. S.: A Course in the Theory of Groups.Graduate Texts in Mathematics 80, Springer, New York (1996). Zbl 0836.20001, MR 1357169, 10.1007/978-1-4419-8594-1 |
| Reference:
|
[7] Wilkens, B.: More on core-2 2-groups.J. Group Theory 20 (2017), 193-225. Zbl 1370.20017, MR 3619126, 10.1515/jgth-2016-0035 |
| . |
