Thread: Element of infinite order

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 = .

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.