Homework Help: Abstract Algebra - Direct Product Question

1. Oct 31, 2006

BSMSMSTMSPHD

I'm supposed to find a non-trivial group G such that G is isomorphic to G x G.

I know G must be infinite, since if G had order n, then G x G would have order n^2. So, after some thought, I came up with the following. Z is isomorphic to Z x Z.

My reasoning is similar to the oft-seen proof that the rationals are countable.

Picture a grid with dots representing each element in Z x Z. Now, starting at the origin, trace a circuitous path (in any direction, but always a tight spiral) and define a map that sends 0 to (0,0), 1 to the next point, -1 to the next point, 2 to the next point, etc.

Is it enough to describe this map in the way I have, or do I need further information (or am I wrong?)

Thanks.

2. Oct 31, 2006

0rthodontist

Z x Z is not isomorphic to Z. Z x Z is not cyclic. You might think about the group of integer functions on the integers.

Last edited: Oct 31, 2006
3. Oct 31, 2006

Hurkyl

Staff Emeritus
That describes a map... (although having only a heuristic description makes it hard to prove things about it)

But you've yet to prove that your map is a homomorphism, that it has an inverse, and that its inverse is a homomorphism.

4. Oct 31, 2006

BSMSMSTMSPHD

Thanks guys. Back to the drawingboard.

5. Oct 31, 2006

Hurkyl

Staff Emeritus
Hrm. I hate to give big hints like this, but...

If a generating set of G must contain at least n elements... then (heuristically speaking) how many elements must a generating set of GxG contain?

6. Oct 31, 2006

BSMSMSTMSPHD

I don't like to give answers like this, but I haven't the slightest idea.

7. Oct 31, 2006

Hurkyl

Staff Emeritus
Well, how many generators does it take to generate the subgroup Gx1 of GxG?

8. Oct 31, 2006

BSMSMSTMSPHD

I would say n - same as for G.

9. Oct 31, 2006

Hurkyl

Staff Emeritus
And what about 1xG? So what does that suggest will be (roughly) true, if you want to generate all of GxG?

10. Oct 31, 2006

BSMSMSTMSPHD

You would need 2n?

11. Nov 1, 2006

Hurkyl

Staff Emeritus
Right. In particular, if G is finitely generated, then...

(this is not a rigorous proof -- I don't know if weird things will happen that allow you to use less than 2n... but we're not looking for proofs here, we're searching for examples!)