Direct sum with isomorphic groups

A follow up to a previous question of mine.

I thought it was true, but according to my book $G \simeq H \oplus K$, with $G \simeq H$ does not imply $K=0$

Is there a simple counter example? In what situation is this true?

(Motivation is, from the previous question, to show that $H_n(X,A)=0$

Edit: The groups are Abelian

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1 Answer

This will be true if the groups are finitely generated, or if they are finite dimensional vector spaces over a field (with the isomorphism being a vector space isomorphism), or in other contexts in which one can appeal to some kind of finite invariant like dimension, free rank, size of the torsion subgroup, etc.

For a counterexample, consider $G = \mathbb Z \oplus \mathbb Z \oplus \cdots$ (countably infinite direct sum). (Exercise: find a suitable choice of $K$.)

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thanks that is great! Does $K=\mathbb{Z}$ work (where $H$ is also a countable infinite direct sum of copies of the integers), or is it more subtle than that... –  Juan S Mar 7 '11 at 6:39
@Qwirk: Yes, this choice of $K$ works! (This is kind of construction/counterexample, based on the equation $\infty + 1 = \infty$, is common in parts of algebra and topolgy.) –  Matt E Mar 7 '11 at 6:45
thanks again –  Juan S Mar 7 '11 at 6:46
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