# Math Help - Element of infinite order

1. ## 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.

2. Suppose $\langle a^i \rangle=\{a^{in}|n \in \mathbb{Z} \}$= $\langle a^j \rangle=\{a^{jn}|n \in \mathbb{Z} \}$.

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.