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 22:58This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question. 


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. 

