# algebra group theory [closed]

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

This 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.

@Rahman: that's a fancy way to ask about $B$ being isomorphic to $B+C$. (Especially in the view of the two answers below :-). – Włodzimierz Holsztyński Jun 23 '13 at 17:50
More interesting question would be: Given $C$, what can we say about groups $B$ such that the direct sum $B+C$ is isomorphic to $B$? – Włodzimierz Holsztyński Jun 23 '13 at 17:57
The post by guest correctly answers the question. This question deserves to be closed now. Please, no changing the question, reinterpreting the question, changing hypotheses, etc.: let OP ask a new question elsewhere if something better comes to mind. – Todd Trimble Jun 23 '13 at 21:16
+1 Todd. Stone soup, and all that – Yemon Choi Jun 23 '13 at 21:24

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 Isaacs 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.

-
For that matter, let A be B direct summed with countably (or more) many copies of C. I think under certain moderate conditions this can characterize such A, but I don't have chapter 5 of "Algebras, Lattices, Varieties" handy. Gerhard "Ask Me About System Design" Paseman, 2013.06.23 – Gerhard Paseman Jun 23 '13 at 18:51

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 to-Hom finite, and $A$ and $B$ are finite). Examples of to-Hom groups are quasicyclic groups and torsion-free groups.
It is not true in general, of course, with the counterexample given in the above answer, where $A$ is a nontrivial group and $G$ the countable direct product of $A$ with itself. Then $G\times A \cong G \times \lbrace e \rbrace$, but $A\neq \lbrace e \rbrace$.

-