A remark on a product of two formational tcc-subgroups

A. A. Trofimuk

Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 495-501

Résumé

A subgroup $A$ of a group $G$ is called tcc-subgroup in $G$, if there is a subgroup $T$ of $G$ such that $G=AT$ and for any $X\le A$ and $Y\le T$ there exists an element $u\in \langle X,Y\rangle $ such that $XY^u\leq G$. The notation $H\le G $ means that $H$ is a subgroup of a group $G$. In this paper we consider a group $G=AB$ such that $A$ and $B$ are tcc-subgroups in $G$. We prove that $G$ belongs to $\mathfrak F$, when $A$ and $B$ belong to $\mathfrak F$ and $\mathfrak F$ is a saturated formation such that $\mathfrak U \subseteq \mathfrak F$. Here $\mathfrak U$ is the formation of all supersoluble groups.

Détail

Keywords: supersoluble group, totally permutable product, saturated formation, tcc-permutable product, tcc-subgroup.