If A, B, C be abelian grops and if A isomorph with direct sum of B and C and A be isomorph with B what we can say about C?

closed as too localized by Todd Trimble♦, Yemon Choi, Bill Johnson, Andy Putman, S. Carnahan♦ Jun 23 '13 at 22:58
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If $A$ is finitely generated then $C$ is necessary trivial by the fundamental theorem of finitely generated abelian groups. In the case in which $A$ is not finitely generated, I like in particular the following counterexample (which you can find in the Isaac`s book ``Algebra. A graduate course"): One can see that $({\mathbb R},+)$ is isomorphic to the direct sum of two copy of itself but, of course, $({\mathbb R},+)$ is not trivial... Anyway, this is not a research level question! 


Nothing, let A be the direct sum of infinite copies of C. Then A is isomorphic to A direct sum C. 


The question is also related to the so called cancellation problem for (not necessarily abelian) groups: If $G\times A\cong G\times B$, when does it follow that $A\cong B$ ? Perhaps it is worth to note that there is a positive answer for finite groups,
and in some other related cases (if $G$ is toHom finite, and $A$ and $B$ are finite).
Examples of toHom groups are quasicyclic groups and torsionfree groups. 

