On a product of two formational $\mathrm{tcc}$-subgroups

A. Trofimuk

Algebra and discrete mathematics, Tome 30 (2020) no. 2, pp. 282-289

Résumé

A subgroup $A$ of a group $G$ is called $\mathrm{tcc}$-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 $\mathrm{tcc}$-subgroups in $G$. We prove that $G$ belongs to $\frak F$, when $A$ and $B$ belong to $\mathfrak F$ and $\mathfrak F$ is a saturated formation of soluble groups 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, $\mathrm{tcc}$-permutable product, $\mathrm{tcc}$-subgroup.