Order of group with three generators

Question

I am trying to work out all the elements of the group generated by three generators and just wanted to know how many elements I would find.

If I have a group $\langle x, y, z \rangle$ with $x^2, y^2$ and $z^2$ equal to the identity, how many elements would this group generate?

Answer 1

If there are no more relations, the elements of the group are the words with letters $x,y,z$ with no consecutive, equal letters. Thus, the group is infinite.