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Problemy Fiziki, Matematiki i Tekhniki (Problems of Physics, Mathematics and Technics), 2022, Issue 1(50), Pages 78–83
DOI: https://doi.org/10.54341/20778708_2022_1_50_78
(Mi pfmt830)
 

This article is cited in 1 scientific paper (total in 1 paper)

MATHEMATICS

On one question of A. N. Skiba in the theory of $\sigma$-properties of finite groups

I. N. Safonova

Belarusian State University, Minsk
Full-text PDF (367 kB) Citations (1)
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Abstract: All considered groups are finite. Let $G$ be a group, $\sigma$ some partition of the set of all primes $\mathbb{P}$, i. e. $\sigma=\{\sigma_i\mid i\in I\}$, where $\mathbb{P}=\bigcup_{i\in I}\sigma_i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\ne j$, $\sigma(G)=\{\sigma_i\mid \sigma_i\cap\pi(|G|)\ne\varnothing\}$. A group $G$ is called $\sigma$-primary if $G$ is a $\sigma_i$-group for some $i=i(G)$. We say that $G$ is a $\sigma$-tower group if either $G=1$ or $G$ has a normal series $1=G_0<G_1<\dots<G_{n-1}<G_n=G$ such that $G_k/G_{k-1}$ is a $\sigma_i$-group, $\sigma_i\in\sigma(G)$, while $G/G_k$ and $G_{k-1}$ are $\sigma_i$-groups for all $k=1,\dots,n$. A subgroup $A$ of $G$ is said to be $\sigma$-subnormal in $G$ if there is a subgroup chain $A=A_0\leqslant A_1\leqslant\dots\leqslant A_t=G$ such that either $A_{i-1}\trianglelefteq A_i$ or $A_i/(A_{i-1})_{A_i}$ is $\sigma$-primary for all $i=1,\dots,t$. In this article, we prove that a non-identity soluble group $G$ is a $\sigma$-tower group if for each $\sigma_i\in\sigma(G)$, where $|\sigma(G)|=n$ a Hall $\sigma_i$-subgroup of $G$ is supersoluble and every $(n+1)$-maximal subgroups of $G$ is $\sigma$-subnormal in $G$. Thus, we give a positive answer to Question 4.8 in [1] in the class of all soluble groups with supersoluble $\sigma$-Hall subgroups.
Keywords: finite group, soluble group, $\sigma$-subnormal subgroup, Sylow tower group, $\sigma$-tower group.
Received: 22.01.2022
Document Type: Article
UDC: 512.542
Language: Russian
Citation: I. N. Safonova, “On one question of A. N. Skiba in the theory of $\sigma$-properties of finite groups”, PFMT, 2022, no. 1(50), 78–83
Citation in format AMSBIB
\Bibitem{Saf22}
\by I.~N.~Safonova
\paper On one question of A.~N.~Skiba in the theory of $\sigma$-properties of finite groups
\jour PFMT
\yr 2022
\issue 1(50)
\pages 78--83
\mathnet{http://mi.mathnet.ru/pfmt830}
\crossref{https://doi.org/10.54341/20778708_2022_1_50_78}
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