Isomorphisms between lattices of nearly normal subgroups

Abstract

A subgroup of a group is said to be nearly normal in if it has a finite index in its normal closure . The set nn(G) of nearly normal subgroups of is a sublattice of the lattice of all subgroups of . Isomorphisms between lattices of nearly normal subgroups of -soluble groups are considered in this paper. In particular, properties of images of normal subgroups under such an isomorphism are investigated. Moreover, it is proved that if is a supersoluble group and is an -soluble group such that the lattices nn(G)and nn(Ḡ) are isomorphic, then also Ḡ is supersoluble.