On S -quasinormal and c -normal subgroups of a finite group

Shirong Li; Yangming Li

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 4, page 1083-1095
  • ISSN: 0011-4642

Abstract

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Let ℱ be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are presented: (1) G ∈ ℱ if and only if there is a normal subgroup H such that G / H ∈ ℱ and every maximal subgroup of all Sylow subgroups of H is either c -normal or S -quasinormally embedded in G . (2) G ∈ ℱ if and only if there is a normal subgroup H such that G / H ∈ ℱ and every maximal subgroup of all Sylow subgroups of F * ( H ) , the generalized Fitting subgroup of H , is either c -normal or S -quasinormally embedded in G . (3) G ∈ ℱ if and only if there is a normal subgroup H such that G / H ∈ ℱ and every cyclic subgroup of F * ( H ) of prime order or order 4 is either c -normal or S -quasinormally embedded in G .

How to cite

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Li, Shirong, and Li, Yangming. "On $S$-quasinormal and $c$-normal subgroups of a finite group." Czechoslovak Mathematical Journal 58.4 (2008): 1083-1095. <http://eudml.org/doc/37887>.

@article{Li2008,
abstract = {Let $\mathcal \{F\}$ be a saturated formation containing the class of supersolvable groups and let $G$ be a finite group. The following theorems are presented: (1) $G\in \mathcal \{F\}$ if and only if there is a normal subgroup $H$ such that $G/H\in \mathcal \{F\}$ and every maximal subgroup of all Sylow subgroups of $H$ is either $c$-normal or $S$-quasinormally embedded in $G$. (2) $G\in \mathcal \{F\}$ if and only if there is a normal subgroup $H$ such that $G/H\in \mathcal \{F\}$ and every maximal subgroup of all Sylow subgroups of $F^*(H)$, the generalized Fitting subgroup of $H$, is either $c$-normal or $S$-quasinormally embedded in $G$. (3) $G\in \mathcal \{F\}$ if and only if there is a normal subgroup $H$ such that $G/H\in \mathcal \{F\}$ and every cyclic subgroup of $F^*(H)$ of prime order or order 4 is either $c$-normal or $S$-quasinormally embedded in $G$.},
author = {Li, Shirong, Li, Yangming},
journal = {Czechoslovak Mathematical Journal},
keywords = {$S$-quasinormally embedded subgroup; $c$-normal subgroup; $p$-nilpotent group; the generalized Fitting subgroup; saturated formation; -nilpotent groups; generalized Fitting subgroup; saturated formations; maximal subgroups; Sylow subgroups; c-normal subgroups; -quasinormally embedded subgroups},
language = {eng},
number = {4},
pages = {1083-1095},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On $S$-quasinormal and $c$-normal subgroups of a finite group},
url = {http://eudml.org/doc/37887},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Li, Shirong
AU - Li, Yangming
TI - On $S$-quasinormal and $c$-normal subgroups of a finite group
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 4
SP - 1083
EP - 1095
AB - Let $\mathcal {F}$ be a saturated formation containing the class of supersolvable groups and let $G$ be a finite group. The following theorems are presented: (1) $G\in \mathcal {F}$ if and only if there is a normal subgroup $H$ such that $G/H\in \mathcal {F}$ and every maximal subgroup of all Sylow subgroups of $H$ is either $c$-normal or $S$-quasinormally embedded in $G$. (2) $G\in \mathcal {F}$ if and only if there is a normal subgroup $H$ such that $G/H\in \mathcal {F}$ and every maximal subgroup of all Sylow subgroups of $F^*(H)$, the generalized Fitting subgroup of $H$, is either $c$-normal or $S$-quasinormally embedded in $G$. (3) $G\in \mathcal {F}$ if and only if there is a normal subgroup $H$ such that $G/H\in \mathcal {F}$ and every cyclic subgroup of $F^*(H)$ of prime order or order 4 is either $c$-normal or $S$-quasinormally embedded in $G$.
LA - eng
KW - $S$-quasinormally embedded subgroup; $c$-normal subgroup; $p$-nilpotent group; the generalized Fitting subgroup; saturated formation; -nilpotent groups; generalized Fitting subgroup; saturated formations; maximal subgroups; Sylow subgroups; c-normal subgroups; -quasinormally embedded subgroups
UR - http://eudml.org/doc/37887
ER -

References

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