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PFMT, 2018, Issue 1(34), Pages 41–44 (Mi pfmt551)  

MATHEMATICS

On finite semi-$p$-decomposable groups

N. M. Adarchenkoa, I. V. Bliznetsa, V. N. Rizhikb

a F. Scorina Gomel State University
b Bryansk State Agrarian University

Abstract: A finite group $G$ is called $p$-decomposable if $G=O_{p'}(G)\times O_p(G)$. We say that a finite group $G$ is semi-$p$-decomposable if the normalizer of every non-normal $p$-decomposable subgroup of $G$ is $p$-decomposable. We prove the following Theorem. Suppose that a finite group $G$ is semi-$p$-decomposable. If a Sylow $p$-subgroup $P$ of $G$ is not normal in $G$, then the following conditions hold: (i) $G$ is $p$-soluble and $G$ has a normal Hall $p'$-subgroup $H$. (ii) $G/F(G)$ is $p$-decomposable. (iii) $O_{p'}(G)\times O_p(G)=H\times Z_\infty(G)$ is a maximal $p$-decomposable subgroup of $G$, and $G/H\times Z_\infty(G)$ is abelian.

Keywords: finite group, $p$-soluble group, $p$-decomposable group, Sylow subgroup, Hall subgroup.

Full text: PDF file (328 kB)
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UDC: 512.542
Received: 26.01.2018

Citation: N. M. Adarchenko, I. V. Bliznets, V. N. Rizhik, “On finite semi-$p$-decomposable groups”, PFMT, 2018, no. 1(34), 41–44

Citation in format AMSBIB
\Bibitem{AdaBliRiz18}
\by N.~M.~Adarchenko, I.~V.~Bliznets, V.~N.~Rizhik
\paper On finite semi-$p$-decomposable groups
\jour PFMT
\yr 2018
\issue 1(34)
\pages 41--44
\mathnet{http://mi.mathnet.ru/pfmt551}


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