Show that Symmetric Group $S3$ and $GL(n,\mathbb F_2)$ is an isomorphism.
$\mathbb F_2$ has $2$ elements - $0,1$.
In $GL(n,\mathbb F_2)$ there are $2$ choices for each entry so $8$ possible matrices in total.
But the symmetric group $3$ has order $6$. The cycles are:
$[],[1,2],[1,3],[2,3][1,2,3],[1,3,2]$
How can I show that they are isomorphic?
First we should remark that the claim is only true for $n=2$. By Cauchy's Theorem, any group of order $2p$ with $p$ prime is either isomorphic to the cyclic group $C_{2p}$, or to the dihedral group $D_p$. For $p=3$ we obtain that there are exactly $2$ non-isomorphic groups of order $6$, namely the cyclic group $C_6$, and the group $D_3=S_3$. Since your group $GL(2,\mathbb{F}_2)$ is not abelian, hence not cyclic, it must be isomorphic to $S_3$.
Edit: I just saw that the question has been answered already; this answer is perhaps also useful, so I will leave it here.