On groups with formational subnormal strictly 2-maximal subgroups

• Дата: 2021
• Автори: Monakhov, V. S., Konovalova , M. N., Монахов , V. S., Коновалова , М. М., Монахов , В. С.
• Формат: Стаття
• Мова: Українська
• Опубліковано: Institute of Mathematics, NAS of Ukraine 2021
• Онлайн доступ: https://umj.imath.kiev.ua/index.php/umj/article/view/1115
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 512.542 Let $H$ be a subgroup of a finite group $G.$ If $G$ contains a maximal subgroup $M$ such that $H$ is a maximal subgroup in $M,$ then $H$ is called a $2$-maximal subgroup of $G.$ A subgroup $U$ of $G$ is said to be a strictly $2$-maximal subgroup in $G$ if $U$ is a $2$-maximal subgroup of $G$ and $U$ is not a 2-maximal subgroup in any proper subgroup of $G.$ We investigate the finite groups with $\mathfrak X$-subnormal strictly $2$-maximal subgroups for arbitrary subgroup-closed formation $\mathfrak X.$ In such a group, any proper subgroup has a nilpotent $\mathfrak X$-residual.We study in more detail the case where $\mathfrak X= \mathfrak A_1\mathfrak F$ for a subgroup-closed formation $\mathfrak F$ and the case where $\mathfrak X$ is a soluble saturated formation.