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ЖУРНАЛЫ // Algebra and Discrete Mathematics // Архив

Algebra Discrete Math., 2020, том 29, выпуск 1, страницы 33–41 (Mi adm736)

Эта публикация цитируется в 5 статьях

RESEARCH ARTICLE

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

N. M. Adarchenko

Department of Mathematics and Technologies of Programming, Francisk Skorina Gomel State University, Gomel 246019, Belarus

Аннотация: Let $\sigma =\{\sigma_{i} \mid i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. $G$ is said to be $\sigma$-soluble if every chief factor $H/K$ of $G$ is a $\sigma_{i}$-group for some $i=i(H/K)$. A set ${\mathcal H}$ of subgroups of $G$ is said to be a complete Hall $\sigma $-set of $G$ if every member $\ne 1$ of ${\mathcal H}$ is a Hall $\sigma_{i}$-subgroup of $G$ for some $\sigma_{i}\in \sigma $ and ${\mathcal H}$ contains exactly one Hall $\sigma_{i}$-subgroup of $G$ for every $i$ such that $\sigma_{i}\cap \pi (G)\ne \varnothing$. A subgroup $A$ of $G$ is said to be ${\sigma}$-quasinormal or ${\sigma}$-permutable in $G$ if $G$ has a complete Hall $\sigma$-set $\mathcal H$ such that $AH^{x}=H^{x}A$ for all $x\in G$ and all $H\in \mathcal H$. We obtain a new characterization of finite $\sigma$-soluble groups $G$ in which $\sigma$-permutability is a transitive relation in $G$.

Ключевые слова: finite group, $\sigma$-permutable subgroup, $P\sigma T$-group, $\sigma$-soluble group, $\sigma$-nilpotent group.

MSC: 20D10, 20D15, 20D30

Поступила в редакцию: 20.01.2020

Язык публикации: английский

DOI: 10.12958/adm1530



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