Non-existence of universal members in classes of Abelian groups

by Shelah. [Sh:622]
J Group Theory, 2001
We prove that if mu^+< lambda =cf(lambda)< mu^{aleph_0}, then there is no universal reduced torsion free abelian group. Similarly if aleph_0< lambda < 2^{aleph_0} . We also prove that if 2^{aleph_0}< mu^+< lambda =cf(lambda)< mu^{aleph_0}, then there is no universal reduced separable abelian p-group in lambda . (Note: both results fail if lambda = lambda^{aleph_0} or if lambda is strong limit, cf (mu)= aleph_0< mu).


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