On lattice isomorphic of mixed abelian groups

Abstract

1. Introduction. In this paper, we investigate when two mixed nonsplitting abelian groups of torsion free rank one are lattice isomorphic. That is, their lattice of subgroups are isomorphic. This is the only outstanding case in the quest to know when abelian groups are lattice isomorphic, as we described in [4]. In [4], we also derived the necessary and sufficient conditions for mixed splitting abelian groups of torsion free rank one to be lattice isomorphic. Ostendorf has obtained a similar result in [6]. These conditions are that the torsion subgroups be isomorphic and the type of one of the groups can be obtained from the type of other by a permutation of the primes, that fixes primes occurring as orders of elements. This generalizes well-known results on when two splitting abelian groups of torsion free rank one are isomorphic. In the 1960's Rotman [8] and Megibben [5] proved that for many classes of mixed abelian groups G of torsion free rank one (eg : countable G) the height matrix U (G) was the distinguishing characteristic. It is our purpose here to extend our results in [4] to obtain parallel generalization of Rotman and Megibben's work. The necessary and sufficient conditions are summarized in the following two Theorems. Theorem I. Let G be a mixed abelian group of torsion free rank one, and assume that G is lattice isomorphic to an abelian group H. Then; H is a mixed abelian group of torsion free rank one, T(G) is isomorphic to T(H), and U(H) can be obtained from U(G) by permutation 11 of primes, with primes occurring as orders of elements of T(G) ~- T(H) fixed. Theorem II. Let G and H be two mixed abelian groups of torsion free rank one with T(G) ~- T(H). And assume that U (H) can be obtained from U (G) by a permutation 17 of primes, which fixes the primes occurring as orders of elements of T(G) ~- T(H). Now if; (i) G and H are splitting over their torsion subgroup, or (ii) G and 1t are countable, or (iii) T(G) ~ T(H) is a direct sum of countable groups, or (iv) T(G) ~ T(H) are both closed, then G is lattice isomorphic to H. We conclude this paper with an example of two mixed abelian groups of torsion free rank one with the same torsion parts and the same height matrices which are not lattice