New characterizations of finite soluble groups

V. A. Vasilyev ; A. N. Skiba

Problemy fiziki, matematiki i tehniki, no. 2 (2010), pp. 28-33

Résumé

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 $G$.

Détail

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