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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, Volume 25, Number 4, Pages 155–163
DOI: https://doi.org/10.21538/0134-4889-2019-25-4-155-163
(Mi timm1681)
 

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

Finite groups with supersoluble subgroups of given orders

V. S. Monakhova, V. N. Tyutyanovb

a Gomel State University named after Francisk Skorina
b Gomel Branch of International University "MITSO"
Full-text PDF (200 kB) Citations (1)
References:
Abstract: We study a finite group $G$ with the following property: for any of its maximal subgroups $H$, there exists a subgroup $H_1$ such that $|H_1|=|H|$ and $H_1\in \frak F$, where $\frak F$ is the formation of all nilpotent groups or all supersoluble groups. We prove that, if $\frak F=\frak N$ is the formation of all nilpotent groups and $G$ is nonnilpotent, then $|\pi (G)|=2$ and $G$ has a normal Sylow subgroup. For the formation $\frak F=\frak U$ of all supersoluble groups and a soluble group $G$ with the above property, we prove that $G$ is supersoluble, or $2\le |\pi (G)|\le 3$; if $|\pi (G)|=3$, then $G$ has a Sylow tower of supersoluble type; if $|\pi (G)|=2$, then either $G$ has a normal Sylow subgroup or, for the largest $p\in \pi (G)$, some maximal subgroup of a Sylow $p$-subgroup is normal in $G$. If $G$ is nonsoluble and, for each maximal subgroup of $G$, there exists a supersoluble subgroup of the same order, then every nonabelian composition factor of $G$ is isomorphic to $PSL_2(p)$ for some prime $p$; we list all such values of $p$.
Keywords: finite group, soluble group, maximal subgroup, nilpotent subgroup, supersoluble subgroup.
Received: 15.04.2019
Revised: 27.06.2019
Accepted: 08.07.2019
Bibliographic databases:
Document Type: Article
UDC: 512.542
MSC: 20D10, 20D20, 20E28
Language: Russian
Citation: V. S. Monakhov, V. N. Tyutyanov, “Finite groups with supersoluble subgroups of given orders”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 4, 2019, 155–163
Citation in format AMSBIB
\Bibitem{MonTyu19}
\by V.~S.~Monakhov, V.~N.~Tyutyanov
\paper Finite groups with supersoluble subgroups of given orders
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2019
\vol 25
\issue 4
\pages 155--163
\mathnet{http://mi.mathnet.ru/timm1681}
\crossref{https://doi.org/10.21538/0134-4889-2019-25-4-155-163}
\elib{https://elibrary.ru/item.asp?id=41455532}
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  • https://www.mathnet.ru/eng/timm/v25/i4/p155
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Trudy Instituta Matematiki i Mekhaniki UrO RAN
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