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PFMT, 2017, Issue 3(32), Pages 36–42 (Mi pfmt515)  

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

MATHEMATICS

On the intersections of maximal subgroups of finite groups containing formation radicals

L. M. Belokon

Mogilev State University of Food Technologies

Abstract: For nonempty radical formation $\mathfrak{F}$ and a finite group $G$ the following statement was proved: if there exist maximal subgroups of $G$ containing $G_{\mathfrak{F}}$, but not containing $G_{\mathfrak{FN}}$, that is $\Phi_{G_{\mathfrak{F}},\overline{G_{\mathfrak{FN}}}}(G)\ne G$, and the factor group $\tilde{\mathrm{F}}_{\Phi_{G_{\mathfrak{F}}}}(G)\cap \Phi_{G_{\mathfrak{F}},\overline{G_{\mathfrak{FN}}}}(G)/\Phi_{G_{\mathfrak{F}}}(G)$ is solvable, then $\Phi_{G_{\mathfrak{F}}}(G)=\Phi_{G_{\mathfrak{F}},\overline{G_{\mathfrak{FN}}}}(G)\subset G_{\mathfrak{FN}}\subseteq\mathrm{F}_{\Phi_{G_{\mathfrak{F}}}}(G)$. In particular, if $G\ne G_{\mathfrak{F}}$ and $\mathrm{Soc}(G/\Phi_{G_{\mathfrak{F}}}(G))=\tilde{\mathrm{F}}_{\Phi_{G_{\mathfrak{F}}}}(G)/\Phi_{G_{\mathfrak{F}}}(G)$ is solvable, then $\Phi_{G_{\mathfrak{F}}}(G)=\Phi_{G_{\mathfrak{F}},\overline{G_{\mathfrak{FN}}}}(G)\subset G_{\mathfrak{FN}}=\tilde{\mathrm{F}}_{\Phi_{G_{\mathfrak{F}}}}(G)$. The corresponding consequences were obtained for products of non-empty radical formations, in particular for $\mathfrak{F}=\mathfrak{N}^{n-1}$, $n$ is any natural number.

Keywords: radical formations of finite groups, products of radical formations, $\mathfrak{F}$-radicals, intersections of maximal subgroups.

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UDC: 512.542
Received: 01.06.2017

Citation: L. M. Belokon, “On the intersections of maximal subgroups of finite groups containing formation radicals”, PFMT, 2017, no. 3(32), 36–42

Citation in format AMSBIB
\Bibitem{Bel17}
\by L.~M.~Belokon
\paper On the intersections of maximal subgroups of finite groups containing formation radicals
\jour PFMT
\yr 2017
\issue 3(32)
\pages 36--42
\mathnet{http://mi.mathnet.ru/pfmt515}


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    This publication is cited in the following articles:
    1. L. M. Belokon, “Peresecheniya maksimalnykh $\theta$-podgrupp, soderzhaschikh formatsionnye radikaly”, PFMT, 2018, no. 1(34), 45–54  mathnet
  • Проблемы физики, математики и техники
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