Finite groups with weakly subnormal Schmidt subgroups in some maximal subgroups

E. V. Zubei

Problemy fiziki, matematiki i tehniki, no. 3 (2022), pp. 82-85.

Résumé

A subgroup $H$ is called weakly subnormal in $G$ if $H=$ for some subgroup $A$ subnormal in $G$ and seminormal subgroup $B$ of $G$. Here the subgroup $B$ is called seminormal in $G$, if there exists a subgroup $Y$ such that $G=BY$ and $BX$ is a subgroup for each subgroup $X$ of $Y$. Finite non-nilpotent group, whose all proper subgroups are nilpotent are called Schmidt. If in a group with a nilpotent maximal subgroup the derived subgroup of a Sylow $2$-subgroup from a maximal subgroup is contained in the center of a Sylow $2$-subgroup, then the group is solvable. If the maximal subgroup of a group is non-nilpotent, then in it there is a Schmidt subgroup. The structure of the group itself, in particular, its solvability depends on the properties of Schmidt subgroups from a maximal subgroup of the group. In this paper, we establish the solubility of a finite group under the condition that some Schmidt subgroups from the maximal subgroup groups are weakly subnormal in a group.

Détail

Keywords: finite group, Schmidt subgroup, weakly subnormal subgroup, maximal subgroup.
Mots-clés : solvable group

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