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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"
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
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
Linking options:
https://www.mathnet.ru/eng/timm1681 https://www.mathnet.ru/eng/timm/v25/i4/p155
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Abstract page: | 227 | Full-text PDF : | 65 | References: | 29 | First page: | 4 |
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