I’m out of the house at the moment but I’ll give some tips to get you started.

1. Show that the elements of $P$ form a group under addition. (if it’s not given that it’s a group). $GL(2 , \mathbb{F})$ forms a group under matrix multiplication.

2. To show $\phi$ is a group homomorphism, show that for any two elements $p, q \in P$

$\phi (p+q) = \phi(p) \cdot \phi(q)$

3. Show that each element of $GL$ is mapped to, keep in mind that $P$ defines that $dg-ef \neq 0$ remember the meaning of that.

4. Keeping in mind that all elements of $GL$ are invertible, what would the null element be?