Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $G, H,$ and $K$ be finitely generated abelian groups. If $G \times K \cong H \times K$, show that $G \cong H$. Show by example that this need not be true if we do not assume that the groups are finitely generated.

I have proved the first requirement, but i have no idea about example. Could anyone help me please? thanks in advance.

share|cite|improve this question
up vote 1 down vote accepted

$$\prod_{i=1}^\infty \Bbb Z\times(\mathbb{Z}\times\Bbb Z)\cong \prod_{i=1}^\infty\Bbb Z\times(\Bbb Z).$$Clearly $\Bbb Z\times\Bbb Z\not\cong\Bbb Z$, hence it isn't necessarily true if they aren't finitely generated.

share|cite|improve this answer

Somewhat more general, let $A\not\cong B$. Then

$$A \times \prod_{i=1}^\infty (B\times A)\cong B \times \prod_{i=1}^\infty (A\times B). $$ since an isomorphism is given by $(a_1,b_1,a_2,b_2,\dots) \mapsto (b_1,a_1,b_2,a_2,\dots)$.

So if we denote $K := \prod_{i=1}^\infty (B \times A)\cong \prod_{i=1}^\infty (B \times A)$, then $A\times K \cong B\times K$.

(This is a pretty standard idea, see for instance

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.