Are the only generators of a Cyclic Group $G=\langle g\rangle$, where $|g| = \infty$, $g$ and $g^{-1}$?

Question

I'm self studying group theory, and this is a question in the textbook I've taken out, there is no answer given so I'm assuming that's because it's too simple to require one. I'm almost certain that the only generators are $g$ and $g^{-1}$, because the group is not finite, so for no $g^n$ can, say, ${(g^n)}^q = 1$, correct?

Answer 1

Hint:

If $g^n$ is another generator, then for some $k$ you can get $g^{nk} = g$. Can $G$ be infinite then?

Answer 2

Any infinite cyclic group is isomorphic to Z And Z has only two generators namely 1 and -1