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