Isomorphisms between lattices of nearly normal subgroups

Abstract

A subgroup $H$ of a group $G$ is said to be nearly normal in $G$ if it has a finite index in its normal closure $H^G$. The set nn(G) of nearly normal subgroups of $G$ is a sublattice of the lattice of all subgroups of $G$. Isomorphisms between lattices of nearly normal subgroups of $FC$-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 $G$ is a supersoluble group and $Ḡ$ is an $FC$-soluble group such that the lattices nn(G)and nn(Ḡ) are isomorphic, then also Ḡ is supersoluble.