I've been studying some abstract algebra, but I don't know how to approach this problem:
Suppose $q$ is a prime power, and let $l$ be a prime such that $l$ divides $q - 1.$ Show that $[F^\times_q : F^{\times l}_q] = m$ where $m \cdot l = q - 1.$
I have no idea how to approach this question. I tried some examples, namely $\mathbb{Z}_7$ and saw that if we look at the multiplicative subgroup $\mathbb{Z}^\times_7$ under the image of $\phi$
$$\phi(x) = x^3$$
we see that
$$\phi(\mathbb{Z}^\times_7) = \{1,6\}.$$
Leaving the world of concrete examples, we can note that: we have at least two options for our order of the subgroup: $m$ and $l$ which follows by Lagrange's Theorem. Aside from this, I don't know where to begin.