| Title:
|
Finite groups whose set of numbers of subgroups of possible order has exactly 2 elements (English) |
| Author:
|
Shao, Changguo |
| Author:
|
Jiang, Qinhui |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
3 |
| Year:
|
2014 |
| Pages:
|
827-831 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Counting subgroups of finite groups is one of the most important topics in finite group theory. We classify the finite non-nilpotent groups $G$ whose set of numbers of subgroups of possible orders $n(G)$ has exactly two elements. We show that if $G$ is a non-nilpotent group whose set of numbers of subgroups of possible orders has exactly 2 elements, then $G$ has a normal Sylow subgroup of prime order and $G $ is solvable. Moreover, as an application we give a detailed description of non-nilpotent groups with $n(G)=\{1, q+1\}$ for some prime $q$. In particular, $G$ is supersolvable under this condition. (English) |
| Keyword:
|
finite group |
| Keyword:
|
number of subgroups of possible orders |
| MSC:
|
20E07 |
| MSC:
|
20E45 |
| idZBL:
|
Zbl 06391528 |
| idMR:
|
MR3298563 |
| DOI:
|
10.1007/s10587-014-0135-4 |
| . |
| Date available:
|
2014-12-19T16:15:59Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144061 |
| . |
| Reference:
|
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| Reference:
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| . |
