Criterions of supersolubility of some finite factorizable groups

Helena V. Legchekova

Algebra and discrete mathematics, no. 3 (2005), pp. 46-55.

Résumé

Let $A$, $B$ be subgroups of a group $G$ and $\mathrm{\varnothing }\ne X\subseteq G$. A subgroup $A$ is said to be $X$-permutable with $B$ if for some $x\in X$ we have $A{B}^x=B^{x}A$ [1]. We obtain some new criterions for supersolubility of a finite group $G=AB$, where $A$ and $B$ are supersoluble groups. In particular, we prove that a finite group $G=AB$ is supersoluble provided $A$, $B$ are supersolube subgroups of $G$ such that every primary cyclic subgroup of $A$ $X$-permutes with every Sylow subgroup of $B$ and if in return every primary cyclic subgroup of $B$ $X$-permutes with every Sylow subgroup of $A$ where $X=F(G)$ is the Fitting subgroup of $G$.

Détail

Mots-clés : finite group, supersoluble group, permutable subgroups, product of subgroups.