On finite groups close to completely factorisable groups

V. A. Vedernikov ; G. V. Savicheva

Diskretnaya Matematika, Tome 19 (2007) no. 2, pp. 78-84

Résumé

A subgroup $H$ of a group $G$ is called complemented in $G$ if a subgroup $K$ exists in $G$ such that $G=HK$ and $H\cap K=1$. A group is called completely factorisable if each subgroup of the group is complemented. Let $D(G)$ be the subgroup of a group $G$ generated by all subgroups of $G$ which have no complements in $G$, $Z(G)$ be the centre of the group $G$, and $\Phi(G)$ be the Frattini subgroup of the group $G$. If all subgroups of $G$ are complemented in $G$, then we set $D(G)=1$. Each cyclic subgroup of the Frattini subgroup $\Phi(G)$ of the group $G$ has no complement in $G$, therefore $\Phi(G)\subseteq D(G)$. In the paper, we obtain a complete description of the structure of a finite group $G$ such that $D(G)\subseteq Z(G)\Phi(G)$.