On some classes of sublattices of the subgroup lattice

Abstract

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 S of normal sections of G a stratification of G, if S contains every trivial normal section of G, and we say that a stratification S of G is G-closed, if S contains every such a normal section of G, which is G-isomorphic to some normal section of G belonging S. Now let S 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 S. In this paper we describe the conditions on S 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.