Let a be a group element that has infinite order. Prove that <a^i> = <a^j> if and only if i = +j or -j.
Suppose $\displaystyle \langle a^i \rangle=\{a^{in}|n \in \mathbb{Z} \}$=$\displaystyle \langle a^j \rangle=\{a^{jn}|n \in \mathbb{Z} \}$.
But if these sets are to be equal, then there must be an integer n such that ni=j and there must be an integer m such that mj=i. That is i|j and j|i. This proves j=+/-i as desired.
The converse is clear, and i trust you can take care of that.