Commutativity Criterions Using Normal Subgroup Lattices

Abstract

We prove that a group $G$ is Abelian whenever (1) it is nilpotent and the lattice of normal subgroups of $G$ is isomorphic to the subgroup lattice of an Abelian group or (2) there exists a non-torsion Abelian group $B$ such that the normal subgroup lattice of $B\times G$ is isomorphic to the subgroup lattice of an Abelian group. Using (2), it is proved that an Abelian group $A$ can be determined in the class of all groups by the lattice of all normal subgroups of some groups, e.g. if $A$ is an Abelian group and $G$ is a group such that $\mathbb{Z}\times A$ and $\mathbb{Z}\times G$ have isomorphic normal subgroup lattices then $A$ and $G$ are isomorphic groups.