Commutativity Criterions Using Normal Subgroup Lattices

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 × 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 × A and \mathbb Z × G have isomorphic normal subgroup lattices then A and G are isomorphic groups.