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 PFMT, 2019, Issue 2(39), Pages 88–91 (Mi pfmt643)

MATHEMATICS

On finite semi-$\pi$-special groups

N. S. Kosenoka, V. M. Selkinb, V. N. Mitsikb, V. N. Rizhikc

a Belarusian Trade and Economic University of Consumer Cooperatives
b F. Scorina Gomel State University
c Bryansk State Agrarian University

Abstract: A finite group $G$ is called $\pi$-special if $G=O_{p_1}(G)\times…\times O_{p_n}(G)\times O_{\pi'}(G)$, where $\pi=\{p_1,…, p_n\}$. We say that a finite group $G$ is semi-$\pi$-special if the normalizer of every non-normal $\pi$-special subgroup of $G$ is $\pi$-special. We prove that if $G$ is not $\pi$-special but $N_G(A)$ is $\pi$-special for every subgroup $A$ of $G$ such that $A$ is either a $\pi'$-group or a $p$-group for some $p\in\pi$, then the following statements hold: (i) $G/F(G)$ is $\pi$-special. Hence $G$ has a Hall $\pi'$-subgroup $H$ and a soluble Hall $\pi$-subgroup $E$. (ii) If $G$ is not $p$-closed for each $p\in\pi$, then: (1) $H$ is normal in $G$ and $E$ is nilpotent. (2) $O_{p_1}(G)\times…\times O_{p_n}(G)\times H$ is a maximal $\pi$-special subgroup of $G$ and every minimal normal subgroup of $G$ is contained in $F(G)$.

Keywords: finite group, $\pi$-soluble group, $\pi$-special group, Sylow subgroup, Hall subgroup.

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Citation: N. S. Kosenok, V. M. Selkin, V. N. Mitsik, V. N. Rizhik, “On finite semi-$\pi$-special groups”, PFMT, 2019, no. 2(39), 88–91

Citation in format AMSBIB
\Bibitem{KosSelMit19} \by N.~S.~Kosenok, V.~M.~Selkin, V.~N.~Mitsik, V.~N.~Rizhik \paper On finite semi-$\pi$-special groups \jour PFMT \yr 2019 \issue 2(39) \pages 88--91 \mathnet{http://mi.mathnet.ru/pfmt643}