A characterization of finite $\sigma$-soluble $P\sigma T$-groups

Chenchen Cao; Dein Wong; Chi Zhang

doi:10.4171/rsmup/108

A characterization of finite

σ

-soluble

P σ T

-groups

Chenchen Cao ; Dein Wong ; Chi Zhang

Rendiconti del Seminario Matematico della Università di Padova, Tome 148 (2022), pp. 203-212.

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Résumé

Let $𝔉$ be a non-empty class of groups, $G$ a finite group and $ℒ (G)$ be the lattice of all subgroups of $G$ . A chief factor $H / K$ of $G$ is $𝔉$ -central in $G$ if $(H / K) ⋊ (G / C_{G} (H / K)) \in 𝔉$ . Let $ℒ_{c 𝔉} (G)$ be the set of subgroups $A$ of $G$ such that every chief factor of $G$ between $A^{G}$ and $A_{G}$ is $𝔉$ -central in $G$ ; let $ℒ_{𝔉} (G)$ be the set of subgroups $A$ of $G$ such that $A^{G} / A_{G} \in 𝔉$ . In this paper, we study the influence of $ℒ_{𝔉} (G)$ and $ℒ_{c 𝔉} (G)$ on the structure of $G$ , where $𝔉$ is a normally hereditary saturated formation containing all $σ$ -nilpotent groups and $D = G^{𝔉}$ is $σ$ -soluble. Moreover, we give a new characterization of a finite $σ$ -soluble group to be a $P σ T$ -group.

Accepté le : 2020-11-25
Publié le : 2022-12-20

MR Zbl

DOI : 10.4171/rsmup/108

Classification : 20

@article{RSMUP_2022__148__203_0,
     author = {Chenchen Cao and Dein Wong and Chi Zhang},
     title = {A characterization of finite $\sigma$-soluble $P\sigma T$-groups},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {203--212},
     volume = {148},
     year = {2022},
     doi = {10.4171/rsmup/108},
     mrnumber = {4542378},
     zbl = {1511.20064},
     language = {en},
     url = {http://www.numdam.org/articles/10.4171/rsmup/108/}
}

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Chenchen Cao; Dein Wong; Chi Zhang. A characterization of finite $\sigma$-soluble $P\sigma T$-groups. Rendiconti del Seminario Matematico della Università di Padova, Tome 148 (2022), pp. 203-212. doi : 10.4171/rsmup/108. http://www.numdam.org/articles/10.4171/rsmup/108/

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