### The Kaplansky test problems for $\aleph_1$-separable groups

by Eklof and Shelah. [EkSh:625]

Proc American Math Soc, 1998

We answer a long-standing open question by proving in ordinary
set theory, ZFC, that the Kaplansky test problems have negative
answers for aleph_1-separable abelian groups of cardinality
aleph_1 . In fact, there is an aleph_1-separable abelian group
M such that M is isomorphic to M oplus M oplus M but not to
M oplus M .

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