
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 Received: 01.03.2010
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 2833
\mathnet{http://mi.mathnet.ru/pfmt160}
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http://mi.mathnet.ru/eng/pfmt160 http://mi.mathnet.ru/eng/pfmt/y2010/i2/p28
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