Finite groups with restrictions on two maximal subgroups

V. S. Monakhov ; A. A. Trofimuk ; E. V. Zubei

Problemy fiziki, matematiki i tehniki, no. 3 (2019), pp. 88-92

Résumé

A subgroup $A$ of a group $G$ is called seminormal in $G$, if there exists a subgroup $B$ such that $G = AB$ and $AB_1$ is a proper subgroup of $G$ for every proper subgroup $B_1$ of $B$. We introduce the new concept that unites subnormality and seminormality. A subgroup $A$ of a group $G$ is called semisubnormal in $G$, if either $A$ is subnormal in $G$, or is seminormal in $G$. In this paper we proved the supersolubility of a group $G$ under the condition that all Sylow subgroups of two non-conjugate maximal subgroups of $G$ are semisubnormal in $G$. Also we obtained the nilpotency of the second derived subgroup $(G')'$ of a group $G$ under the condition that all maximal subgroups of two non-conjugate maximal subgroups are semisubnormal in $G$.

Détail

Keywords: supersoluble groups, semisubnormal subgroup, derived subgroup, Sylow subgroup, maximal subgroup.