# 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 ?