On some classes of sublattices of the subgroup lattice

In this paper $G$ always denotes a group. If $K$ and $H$ are subgroups of $G$, where $K$ is a normal subgroup of $H$, then the factor group of $H$ by $K$ is called a section of $G$. Such a section is called normal, if $K$ and $H$ are normal subgroups of $G$, and trivial, if $K$ and $H$ are equal. We call any set $\Sigma$ of normal sections of $G$ a stratification of $G$, if $\Sigma$ contains every trivial normal section of $G$, and we say that a stratification $\Sigma$ of $G$ is $G$-closed, if $\Sigma$ contains every such a normal section of $G$, which is $G$-isomorphic to some normal section of $G$ belonging $\Sigma$. Now let $\Sigma$ be any $G$-closed stratification of $G$, and let $L$ be the set of all subgroups $A$ of $G$ such that the factor group of $V$ by $W$, where $V$ is the normal closure of $A$ in $G$ and $W$ is the normal core of $A$ in $G$, belongs to $\Sigma$. In this paper we describe the conditions on $\Sigma$ under which the set $L$ is a sublattice of the lattice of all subgroups of $G$ and we also discuss some applications of this sublattice in the theory of generalized finite $T$-groups.