The Kaplansky test problems for aleph₁-separable groups

الأصل · EN

We answer a long-standing open question by proving in ordinary set theory, ZFC, that the Kaplansky test problems have negative answers for aleph₁-separable abelian groups of cardinality aleph₁. In fact, there is an aleph₁-separable abelian group M such that M is isomorphic to M oplus M oplus M but not to M oplus M.