On lattice-isomorphic Abelian groups
The authors consider the question of when two Abelian groups of torsion-free rank one can have the same lattice of subgroups. They review the history of published, sometimes incorrect, results on lattice-isomorphic Abelian groups, and carefully derive when two splitting Abelian groups of torsion-free rank one can be lattice-isomorphic. This occurs precisely when their torsion subgroups are isomorphic and the type of one of the groups can be obtained from the type of the other by some permutation of the primes which leaves fixed all primes occurring as orders of elements. The method of proof involves a direct construction of the correspondence between cyclic subgroups and hence all subgroups.
scientific article; zbMATH DE number 62260
scientific article