Equality of two sets

Let $p$, $l$ two prime numbers such that $lmid p-1$. Let $mathbb{P}$ the subgroup of $mathbb{Z}_{p}^{*}$ of order $l$ and $mathbb{L}$ the subgroup of $mathbb{Z}_{p}^{*}$ of order $frac{p-1}{l}$.

Assuming that $l^2$ $nmid$ $p-1$. I need to proof $psi(mathbb{Z}_{p}^{*})=mathbb{L}$ where
$psi$ is the homeomorphism:

$psi:mathbb{Z}_{p}^{*}tomathbb{Z}_{p}^{*}$, $xmapsto x^l$.

I proof that $psi(mathbb{Z}_{p}^{*})subset mathbb{L}$.
But I don’t know how to proof $mathbb{L}subsetpsi(mathbb{Z}_{p}^{*})$.

I think problem can be resume like:
Let $y in mathbb{L}$, proof that it exists an $xin mathbb{Z}_{p}^{*}$ such that $y=x^l$

But I don’t know how to proof it.

I also need to proof that $y$ have $l$ antecedents.
Can someone help me please ?