Cosets of $\Bbb Z/p$ in $\Bbb Z/p^2$: do $g, 2g, 3g, \ldots, pg$ lie in separate cosets where $\langle g \rangle = \Bbb Z/p^2$?

Question

Consider the short exact sequence:

$$0 \to \Bbb Z/p \to \Bbb Z/p^2 \to \Bbb Z/p \to 0$$

I'm trying to analyze the cosets of $\Bbb Z/p$ in $\Bbb Z/p^2$. If $g$ is a generator of $\Bbb Z/p^2$, is it true that $g, 2g, 3g, \ldots, pg$ lie in separate cosets? I tried out a few examples, like the case of $p = 5$ and it seems true, but I'm not sure how to prove it.