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 PFMT, 2018, Issue 4(37), Pages 103–105 (Mi pfmt612)

MATHEMATICS

On the generalized norm of a finite group

V. M. Selkina, N. S. Kosenokb

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

Abstract: Let $G$ be a finite group and $\pi=\{p_1,…,p_n\}\subseteq\mathbb{P}$. Then $G$ is called $\pi$-special if $G=O_{p_1}(G)\times…\times O_{p_n}(G)\times O_{\pi'}(G)$. We use $\mathfrak{N}_{\pi sp}$ to denote the class of all finite $\pi$-special groups. Let $\mathrm{N}_{\pi sp}$ be the intersection of the normalizers of the $\pi$-special residuals of all subgroups of $G$, that is, $\mathrm{N}_{\pi sp}(G)=\bigcap\limits_{H\leqslant G}N_G(H^{\mathfrak{N}_{\pi sp}})$. We say that $\mathrm{N}_{\pi sp}$ is the $\pi$-special norm of $G$. We study the basic properties of the $\pi$-special norm of $G$. In particular, we prove that $\mathrm{N}_{\pi sp}$ is $\pi$-soluble.

Keywords: finite group, $\pi$-special group, $\pi$-soluble group, $\pi$-special residual of a group, $\pi$-special norm of a group.

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Citation: V. M. Selkin, N. S. Kosenok, “On the generalized norm of a finite group”, PFMT, 2018, no. 4(37), 103–105

Citation in format AMSBIB
\Bibitem{SelKos18} \by V.~M.~Selkin, N.~S.~Kosenok \paper On the generalized norm of a finite group \jour PFMT \yr 2018 \issue 4(37) \pages 103--105 \mathnet{http://mi.mathnet.ru/pfmt612}