Are the sylow subgroups of multiplicative group of a finite field unique?

Question

Let $\mathbb{F}_q$ be a finite field with $q=p^w$ elements for some prime $p$.

Let $l$ be another prime which divides $q-1$.

My question is if there is a unique sylow $l$-subgroup of $\mathbb{F}_q^*$.

Here is my proof attempt:

Write $q-1$ as $q-1=l^el'$ where $\gcd(l,l')=1$.

Sylow Theorem 1 guarantee's that there exists a sylow $l$-subgroup of order $l^e$. Furthurmore, it is cyclic and generated by an element $a$ of order $l^e$ because $\mathbb{F}_q^*$ is cyclic.

Sylow Theorem 2 says that all the sylow $l$-subgroups are conjugate. Since $\mathbb{F}_q^*$ is abelian, they are all equal, hence there is only one.

Answer 1

You are correct ......................................