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On the freeness of abelian groups: A generalization of Pontryagin's theorem

1970
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Bulletin of the American Mathematical Society
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Pontryagin gave an example that demonstrates that the prefix "/o-" cannot be deleted from the above theorem; indeed he showed that there exists a torsion-free group of rank 2 that is not free such that each subgroup of rank 1 is free (see, for example, [l, p. 151 ]). We present the following direct generalization of Pontryagin's theorem obtained by transposing the countability condition. THEOREM 1. If the torsion-free abelian group G is the fa-union of a countable number of pure subgroups that

doi:10.1090/s0002-9904-1970-12586-1
fatcat:b5d5rehcljfm3f2c3pdehgopby