I am given the set $W$ of 1-dimensional vector subspaces in the $\mathbb{F_3}$-vector space $\mathbb{F_3^2}$. (We know that |$W$|$=4$).
The group $G:=GL_2(\mathbb{F_3})$ actson the set $W$ in the following way:
$$ g·V := \{g·w | w \in V \} \in W (g \in G, V \in W) $$
I need to show:
(a) This operation induces a group homomorphism $f: G - S_4$.
(b) Determine the $ker(f), im(f)$ and the order of G.
I have been able to prove (a), via the homomorphism defined as follows:
\begin{align*} f: G&\rightarrow S_4\\ g&\mapsto \pi_g \end{align*}, where \begin{align*} \pi_g: W&\rightarrow W\\ w&\mapsto g·w \end{align*}
Also, I've obtained that $ker(f)$ is the intersection of all the stablisators of elements $w \in W$. However, I don't seem to be able to determine $im(f)$ (I know that the order of $G$ is 48).
There is a further question:
(c) Consider the following subgroups of $G$
$$ N=\{S \in G | det(S)=1, S^2=+-E\}, H=\{\quad \begin{pmatrix} 1 & b \\ 0 & d \end{pmatrix} | b, d \in \mathbb{F_3}; d \neq 0 \} $$
Show that $G=H·N$.
I have barely made progress with (c). I think you may have to use the second isomorphism theorem, but I can't get beyond that.
Any help would be appreciated, thanks a lot.