2

I've asked a similar question: Computing Quotient Groups $\mathbb{Z}_4 \times \mathbb{Z}_{10} / \langle (2, 4) \rangle$, $\mathbb{Z} \times \mathbb{Z}_{6}/ \langle (1, 2) \rangle$

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.

Community
  • 1
Student
  • 3,431

1 Answers1

4

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 $\mathbb{Z}^2\times \mathbb{Z}_3$.

Arturo Magidin
  • 417,286
  • Do you know a good reference where I can get more practice with these kinds of problems? – Student Dec 13 '11 at 06:07
  • 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 06:08