Show that there is homomorphism from $\mathrm{GL}(1,4)$ to $\mathrm{GL}(2,2)$, where $\mathrm{GL}(n,k)$ denotes the general linear group, ie invertible $n\times n$ matrices with entries in $\mathbb{F}_k$, such that it corresponds to the inclusion of $A_3$ in $S_3$ and it is injective. Furthermore, show that the same map is a ring homomorphism from $\mathbb{F}_4$ to all $2 \times 2$ matrices with entries in $\mathbb{F}_2$.
I'm not sure what "corresponds to the inclusion" means? Could someone help me? I just need a map but not sure where to start.
An element $x$ in $\mathrm{GL}(1,4)$ is just a non zero element in $F_4$.
The main idea is to consider $F_4$ as a vector space of dimension 2 over $F_2$.
Then multiplication by $x$ is a $F_2$ linear map $F_4\to F_4$, and therefore an element in $\mathrm{GL}(2,F_2)$. But this application is also defined if $x=0$.
Summing up, the map $\mu :F_4\to L(F_4)$ , where $L(F_4)$ is the ring of $F_2-$linear maps $F_4\to F_4$, defined by $\mu(x).y=x.y$ is a ring homomorphism from $F_4$ to the ring of $F_2-$linear maps $F_4\to F_4$.