The question is to show that $GL_2(\mathbb{F}_2)$ and $S_3$ are isomorphic where $S_3$ is the symmetric group of $\{1, 2, 3\}$ and the group operation is composition.
I have listed all the elements in both groups (there are $6$) and also classified them (self inverse, identity etc.). I have an idea as to which elements correspond to each other but if I specifically define the group homomorphism for each element it will take too long to prove it actually is a homomorphism.
Is there any other way to define a homomorphism between these two groups? I have seen similar questions but they were on permutation matrices which is not the case in this question so I am not exactly sure what to do.
The explicit way is clearly possible and not too long. It will be enough to send generators to generators.
Of course, it is much shorter to note that both groups are of order $6$ and non-abelian, hence isomorphic. This follows from Cauchy's Theorem, for the primes $p=2$ and $p=3$ dividing the group order $6$.
Possible duplicates:
Prove $GL_2(\mathbb{Z}/2\mathbb{Z})$ is isomorphic to $S_3$
Show that Symmetric Group $S3$ and $GL(n,\mathbb F_2)$ is an isomorphism
Find isomorphism between $S_3$ and $GL_2(F_2)$.
An isomorphism from $GL_{2}(\mathbb{F_{2}})$ to $S_{3}$