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 PFMT, 2019, Issue 3(40), Pages 107–110 (Mi pfmt665)

MATHEMATICS

Finite groups with given local sections

B. Hua, J. Huanga, A. N. Skibab

a Jiangsu Normal University, Xuzhou
b F. Scorina Gomel State University

Abstract: A group is called primary if it is a finite $p$-group for some prime $p$. If $\sigma=\{\sigma_i\mid i\in I\}$ is some partition of $\mathbb{P}$, that is, $P=\bigcup_{i\in I}\sigma_i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\ne j$, then we say that a finite group $G$ is: $\sigma$-primary if it is a $\sigma_i$-group for some $i$; $\sigma$-nilpotent if $G=G_1\times…\times G_n$ for some $\sigma$-primary groups $G_1,…,G_n$. If $N=N_G(A)$ for some primary non-identity subgroup $A$ of $G$, then we say that $N/A_G$ is a local section of $G$. In this paper, we study a finite group $G$ under hypothesis that all proper local sections of $G$ belong to a saturated hereditary formation $\mathfrak{F}$, and we determine the normal structure of $G$ in the case when all local sections of $G$ are $\sigma$-nilpotent.

Keywords: finite group, hereditary saturated formation, $\mathfrak{F}$-hypercentre, local section, $\sigma$-nilpotent group.

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Citation: B. Hu, J. Huang, A. N. Skiba, “Finite groups with given local sections”, PFMT, 2019, no. 3(40), 107–110

Citation in format AMSBIB
\Bibitem{HuHuaSki19} \by B.~Hu, J.~Huang, A.~N.~Skiba \paper Finite groups with given local sections \jour PFMT \yr 2019 \issue 3(40) \pages 107--110 \mathnet{http://mi.mathnet.ru/pfmt665}