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.