What is the significance of free products in group theory?

[dumbQuestion]

Hey I am really confused over free products. So I understand abstractly, I think. If we have two groups G, H, then the free product G*H would be the group where elements are finite reduced words of arbitrary length, i.e., powers of elements of g and h, where elements of the same group don't sit next to each other (ex. g^1g^2h^3 is NOT a reduced word because it would be g^3h^3.)

The thing I don't understand then is, if I have say the same group, what would be G*G? Because every combination would just be elements of G which are next to each other. I mean, take for example, g^2g^3. This would reduce to g^5. So aren't I just going to get G again?Like when I think of the free product of Z * Z. How is this not just Z? Cus a word would be just like 2*5*6*... etc. (finite length). Then every single word would reduce.

 

[Monobrow]

I am fairly certain that when we take the free product of a group with itself, we formally think of the elements as being distinct if they come from a different copy of the group. For example, the free product of Z with itself is the free group F_2. It might be useful when doing something like this to mark one copy of the group with a 'dash' e.g write Z*Z' and distinguish the elements of Z' in a similar way.

 

[dumbQuestion]

So what does this look like exactly then? Since the words can be any arbitrary finite length, I can't think of it as having some sort of dimension like I normally would when thinking of say Z X Z. What is this group giving me (Z * Z that is). Just some random combination of integers?

 

[Monobrow]

Z is just an infinite cyclic group. So give your copies of Z presentations as follows: Z=<a|-> and Z=<b|-> (the generators are distinct because formally we think of the two copies of Z as being distinct). Then Z*Z=<a,b|-> by the explanation here: http://en.wikipedia.org/wiki/Free_product under "Presentations".
So Z*Z is just the free group on two generators. There's no point in considering what the individual elements of each Z are (or else we will have an image of a bunch of integers randomly thrown together as you say), just that they are infinite cyclic and that we can treat their two generators as being distinct.

 

[dumbQuestion]

oh ok, so it seems like maybe I was thinking of things backwards. This mechanism of using free groups and free products, presentation, etc., is just a way to describe a group. So maybe its not that Z*Z is just something that I want to use for a calculation, maybe its more like, I have some group G, and if I realize some information about it, I might realize its isomorphic to Z*Z, this makes it easier to deal with, etc. This whole process is just kind of a way of describing groups? Am I thinking in the right direction?