# Question about isomorphic direct products of groups and isomorphic factors.

1. Dec 4, 2012

### IvanT

1. The problem statement, all variables and given/known data
Suppose G and F are groups and GxF is isomorphic to G'xF', if G is isomorphic to G', can we conclude that F is isomorphic to F'?

2. Relevant equations

3. The attempt at a solution
I'm trying to give a proof using the first isomorphism theorem (using that GxF/Gx(e) is isomorphic to F, and that G'xF'/G'x(e) is isomorphic to F'), but I can't find an isomorphism between the quotients. I also can't find a counter example of the statement, so any help or suggestions would be appreciated.

2. Dec 5, 2012

### micromass

Staff Emeritus
Did you know that $\mathbb{R}^2$ and $\mathbb{R}$ are isomorphic as groups?? Try to prove this.

3. Dec 5, 2012

### IvanT

I didn't know that, thanks a lot that solves my problem.

The statement of the problem is false then. Because if RxR is isomorphic to R, then it's also isomorphic to Rx(e), and the statement of the problem would imply that R is isomorphic to the trivial group, which is false.

Thanks : )

4. Dec 5, 2012

### micromass

Staff Emeritus
Yeah, but I think you still need to prove that $\mathbb{R}^2$ is isomorphic to $\mathbb{R}$. This is not trivial.

5. Dec 5, 2012

### IvanT

Yeah, I still need to prove that, but at least I know that the initial statement is wrong.

6. Dec 5, 2012

### Dick

If you want an example that's a little more manageable then the R^2, R thing, try taking G to be an infinite direct product of factors of Z (the integers). Or any other group you like.

7. Dec 5, 2012

### micromass

Staff Emeritus
That one is actually a really nice counterexample, since it generalizes to other fields of mathematics as well. A similar example works in topology, for example. The $\mathbb{R}^2$ thing does not.

8. Dec 5, 2012

### Dick

And you can actually write down what the isomorphism is explicitly.

9. Dec 5, 2012

### IvanT

Thanks a lot, that works.