On σ -permutably embedded subgroups of finite groups

Chenchen Cao; Li Zhang; Wenbin Guo

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 1, page 11-24
  • ISSN: 0011-4642

Abstract

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Let σ = { σ i : i ∈ I } be some partition of the set of all primes ℙ , G be a finite group and σ ( G ) = { σ i : σ i ∩ π ( G ) ≠ ∅ } . A set ℋ of subgroups of G is said to be a complete Hall σ -set of G if every non-identity member of ℋ is a Hall σ i -subgroup of G and ℋ contains exactly one Hall σ i -subgroup of G for every σ i ∈ σ ( G ) . G is said to be σ -full if G possesses a complete Hall σ -set. A subgroup H of G is σ -permutable in G if G possesses a complete Hall σ -set ℋ such that H A x = A x H for all A ∈ ℋ and all x ∈ G . A subgroup H of G is σ -permutably embedded in G if H is σ -full and for every σ i ∈ σ ( H ) , every Hall σ i -subgroup of H is also a Hall σ i -subgroup of some σ -permutable subgroup of G . By using the σ -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.

How to cite

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Cao, Chenchen, Zhang, Li, and Guo, Wenbin. "On $\sigma $-permutably embedded subgroups of finite groups." Czechoslovak Mathematical Journal 69.1 (2019): 11-24. <http://eudml.org/doc/294418>.

@article{Cao2019,
abstract = {Let $\sigma =\lbrace \sigma _i\colon i\in I\rbrace $ be some partition of the set of all primes $\mathbb \{P\}$, $G$ be a finite group and $\sigma (G)=\lbrace \sigma _i\colon \sigma _i\cap \pi (G)\ne \emptyset \rbrace $. 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 $G$. 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.},
author = {Cao, Chenchen, Zhang, Li, Guo, Wenbin},
journal = {Czechoslovak Mathematical Journal},
keywords = {finite group; $\sigma $-subnormal subgroup; $\sigma $-permutably embedded subgroup; $\sigma $-soluble group; supersoluble group},
language = {eng},
number = {1},
pages = {11-24},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On $\sigma $-permutably embedded subgroups of finite groups},
url = {http://eudml.org/doc/294418},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Cao, Chenchen
AU - Zhang, Li
AU - Guo, Wenbin
TI - On $\sigma $-permutably embedded subgroups of finite groups
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 1
SP - 11
EP - 24
AB - Let $\sigma =\lbrace \sigma _i\colon i\in I\rbrace $ be some partition of the set of all primes $\mathbb {P}$, $G$ be a finite group and $\sigma (G)=\lbrace \sigma _i\colon \sigma _i\cap \pi (G)\ne \emptyset \rbrace $. 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 $G$. 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.
LA - eng
KW - finite group; $\sigma $-subnormal subgroup; $\sigma $-permutably embedded subgroup; $\sigma $-soluble group; supersoluble group
UR - http://eudml.org/doc/294418
ER -

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