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

Paul Hill
1970 Bulletin of the American Mathematical Society  
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
more » ... ure subgroups that are free, then G must be free. OUTLINE OF PROOF. Let G be an /tr-union of pure subgroups H nt n For simplicity of notation, let JU denote the smallest ordinal having the cardinality of G. We claim that there exist subgroups A at a<fx, of G satisfying the following conditions: (0) ^o = 0. (1) Aa is pure in G for each a<fx. (2) {AayH n } is pure in G for each a</x and each n<w. (3) Aa+i^Aa for each a such that a + Kju. (4) A a+i/A a is countable for each a such that a +1< ju. (5) A a C\H n = Yliei(n,a){gi} for OJ</X and #<co, where I(n, a) is a subset of I(n). A MS 1969 subject classifications. Primary 2030; Secondary 2051.
doi:10.1090/s0002-9904-1970-12586-1 fatcat:b5d5rehcljfm3f2c3pdehgopby