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17.21 (2010)

Solved

a) If $A, B, C$ are torsion-free abelian groups with $A \cong A \oplus B \oplus C$, must $A \cong A \oplus B$?
b) What if, furthermore, $B \cong C$?

Contributor: G. M. Bergman

Progress

No, not necessarily, even in case b). See (A. L. S. Corner, in Proc. Colloq. Abelian Groups (Tihany, 1963), Budapest, 1964, 43–48); I also have an example with $B$ and $C$ of rank 1 as torsion-free abelian groups. (G. M. Bergman, Letter of 20 February 2011.)

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