Let $p\geq 2$ be a prime and let g be an element of order $p-1$ in $\Bbb Z_p$. Prove that every non-zero element of $\Bbb Z_p$ can be written as a power of $g$.
So i wanted to start this proof by proving the the elements $[g],[g]^2,[g]^3,...,[g]^{p-1}$ are all distinct. But im a bit uncertain on how do this . I thought it could be something with inverses since we are in $\Bbb Z_p$ but that didn't really workout.
Hint Suppose the powers $g, g^2, \ldots, g^{p - 1}$ are not distinct, so that $g^k = g^l$ for some $1 \leq k < l \leq p - 1$. Multiply both sides by $g^{-k}$.