The Formation of Finite Groups with a Supersolvable $\pi$-Hall Subgroup

Yi Xiaolan ; L. A. Shemetkov

Matematičeskie zametki, Tome 87 (2010) no. 2, pp. 280-286

Résumé

Let $G$ be a finite group having a $\pi$-subgroup $H$ such that $|G:H|$ is not divisible by the numbers in $\pi$. In this case, the subgroup $H$ is referred to as a $\pi$-Hall subgroup, and the group $G$ by itself is referred to as a $E_\pi$-group. If, moreover, the group $H$ is supersolvable, then $G$ is referred to as an $E_\pi^u$-group. It is proved in the paper that the class of all $E_\pi^u$-groups is a solvably saturated formation, and the proof is carried out without using the classification of finite groups.

Détail

Keywords: Hall subgroup, classification of finite groups, saturated formation, Frattini subgroup, monolithic group, subdirect product.
Mots-clés : supersolvable group, formation