# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 122, 2009, pp. 161–169**

**DOI: 10.4171/RSMUP/122-10**

Published online: 2009-12-31

Commutativity Criterions Using Normal Subgroup Lattices

Simion Breaz^{[1]}(1) Babes-Bolyai University, Cluj-Napoca, Romania

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.

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Breaz Simion: Commutativity Criterions Using Normal Subgroup Lattices. *Rend. Sem. Mat. Univ. Padova* 122 (2009), 161-169. doi: 10.4171/RSMUP/122-10