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 PFMT, 2010, Issue 2(3), Pages 28–33 (Mi pfmt160)

MATHEMATICS

New characterizations of finite soluble groups

V. A. Vasilyev, A. N. Skiba

F. Skorina Gomel State University, Gomel

Abstract: A subgroup $H$ of a group $G$ is called modular in $G$ if $H$ is a modular element (in sense of Kurosh) of the lattice $L(G)$ of all subgroups of $G$. The subgroup of $H$ generated by all modular subgroups of $G$ contained in $H$ is called the modular core of $H$ and denoted by $H_{mG}$. In the paper, we introduce the following concepts. A subgroup $H$ of a group $G$ is called $m$-supplemented ($m$-subnormal) in $G$ if there exists a subgroup (a subnormal subgroup respectively) $K$ of $G$ such that $G = HK$ and $H \cap K \le H_{mG}$. We proved the following theorems.
Theorem A. A group $G$ is soluble if and only if each Sylow subgroup of $G$ is $m$-supplemented in $G$.
Theorem B. A group $G$ is soluble if and only if every its maximal subgroup is $m$-subnormal in $G$.

Keywords: finite group, soluble group, subnormal subgroup, modular subgroup, modular core, $m$-supplemented subgroup, $m$-subnormal subgroup

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UDC: 512.542

Citation: V. A. Vasilyev, A. N. Skiba, “New characterizations of finite soluble groups”, PFMT, 2010, no. 2(3), 28–33

Citation in format AMSBIB
\Bibitem{VasSki10} \by V.~A.~Vasilyev, A.~N.~Skiba \paper New characterizations of finite soluble groups \jour PFMT \yr 2010 \issue 2(3) \pages 28--33 \mathnet{http://mi.mathnet.ru/pfmt160}