On groups with formational subnormal strictly 2-maximal subgroups
Abstract
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.
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Copyright (c) 2021 Victor Monakhov
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