# Commutativity Criterions Using Normal Subgroup Lattices

### Simion Breaz

Babes-Bolyai University, Cluj-Napoca, Romania

## 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* × *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 *Z* × *A*and *Z* × *G* have isomorphic normal subgroup lattices then *A* and *A* are isomorphic groups.

## Cite this article

Simion Breaz, Commutativity Criterions Using Normal Subgroup Lattices. Rend. Sem. Mat. Univ. Padova 122 (2009), pp. 161–169

DOI 10.4171/RSMUP/122-10