| Title:
|
On $\sigma $-permutably embedded subgroups of finite groups (English) |
| Author:
|
Cao, Chenchen |
| Author:
|
Zhang, Li |
| Author:
|
Guo, Wenbin |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
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0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
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69 |
| Issue:
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1 |
| Year:
|
2019 |
| Pages:
|
11-24 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\sigma =\{\sigma _i\colon i\in I\}$ be some partition of the set of all primes $\mathbb {P}$, $G$ be a finite group and $\sigma (G)=\{\sigma _i\colon \sigma _i\cap \pi (G)\neq \emptyset \}$. A set $\mathcal {H}$ of subgroups of $G$ is said to be a complete Hall $\sigma $-set of $G$ if every non-identity member of $\mathcal {H}$ is a Hall $\sigma _i$-subgroup of $G$ and $\mathcal {H}$ contains exactly one Hall $\sigma _i$-subgroup of $G$ for every $\sigma _i\in \sigma (G)$. $G$ is said to be $\sigma $-full if $G$ possesses a complete Hall $\sigma $-set. A subgroup $H$ of $G$ is $\sigma $-permutable in $G$ if $G$ possesses a complete Hall $\sigma $-set $\mathcal {H}$ such that $HA^x$= $A^xH$ for all $A\in \mathcal {H}$ and all $x\in G$. A subgroup $H$ of $G$ is $\sigma $-permutably embedded in $G$ if $H$ is $\sigma $-full and for every $\sigma _i\in \sigma (H)$, every Hall $\sigma _i$-subgroup of $H$ is also a Hall $\sigma _i$-subgroup of some $\sigma $-permutable subgroup of $G$. \endgraf By using the $\sigma $-permutably embedded subgroups, we establish some new criteria for a group $G$ to be soluble and supersoluble, and also give the conditions under which a normal subgroup of $G$ is hypercyclically embedded. Some known results are generalized. (English) |
| Keyword:
|
finite group |
| Keyword:
|
$\sigma $-subnormal subgroup |
| Keyword:
|
$\sigma $-permutably embedded subgroup |
| Keyword:
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\hbox {$\sigma $-soluble} group |
| Keyword:
|
supersoluble group |
| MSC:
|
20D10 |
| MSC:
|
20D20 |
| MSC:
|
20D35 |
| idZBL:
|
Zbl 07088765 |
| idMR:
|
MR3923570 |
| DOI:
|
10.21136/CMJ.2018.0148-17 |
| . |
| Date available:
|
2019-03-08T14:53:52Z |
| Last updated:
|
2021-04-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147613 |
| . |
| Reference:
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