Show ZXZ is an infinite cyclic group.

[Daveyboy]

Homework Statement

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

Homework Equations

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).

 

[Daveyboy]

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

 

[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.