# Computing Quotient Groups with Infinite Groups

I've asked a similar question: Computing Quotient Groups

But now I want to compute a quotient group involving a direct product in which every direct factor is infinite. For example $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z} / \langle(1, 1, 1)\rangle$ or $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z} / \langle(3, 3, 3)\rangle$. Is there a better approach than just looking for a homomorphism?

Thanks in advance.

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"The direct product of all infinite groups" would be a very big thing indeed. I think you mean "a direct product in which every direct factor is infinite." –  Arturo Magidin Dec 13 '11 at 6:00
@Arturo: Thanks for the suggestion. –  Student Dec 13 '11 at 6:02

## 1 Answer

In the case of a finite direct product of copies of $\mathbb{Z}$, the Smith Normal Form solves the problem for you.

Here, note that $\mathbb{Z}^3$ has a basis of the form $(1,0,0)$, $(1,1,0)$, and $(1,1,1)$, so your first quotient is just isomorphic to $\mathbb{Z}^2$ (you are just "killing" one generator). The same basis tells you that the second quotient is isomorphic to $\mathbb{Z}^2\times \mathbb{Z}_3$.

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Do you know a good reference where I can get more practice with these kinds of problems? –  Student Dec 13 '11 at 6:07
Nothing specific to the kind of problems you seem to be looking at; any book on Abstract Algebra (Lang, Dummit and Foote, Herstein, Jacobson) will have the material and exercises on it, but probably not many. Same with introductory books to group theory (e.g., Rotman). –  Arturo Magidin Dec 13 '11 at 6:08