Show ZXZ is an infinite cyclic group.

1. Jan 11, 2009

Daveyboy

1. The problem statement, all variables and given/known data

Show ZXZ is an infinite cyclic group. Under addition of course.

2. Relevant equations

3. The attempt at a solution
So this obviously is an infinite cyclic group. Z is generated by <1> or <-1>.
The problem I run into here is I think <(1,1)> will only generate elements of the form (a,a) s.t. a is an element of Z, (think along the diagonal). I do not see a way to generate something like (2,3).

2. Jan 11, 2009

Daveyboy

Wow, sorry I was confused, I see that there can not be any generators of ZXZ.

3. Jan 11, 2009

Dick

There can be generators. You just need more than one to generate the group. E.g. {(1,0), (0,1)} is a set that generates the whole group.