Showing that the general linear group is isomorphic to symmetric group.

Question

Let $G = GL(2, \Bbb Z_2)$, the general linear group of $2 \times 2$ invertible matrices with coefficients in $\Bbb Z_2$. Show that $G$ is isomorphic to $S_3$.

I'm having trouble getting started with this one. Both groups have six elements and I know I need to show that there is a bijection $f:G \to S_3$. I don't really know how to go about this. Any help would be appreciated.

Answer 1

The vector space $\mathbb{F}_2^2$ has exactly three nonzero elements. So, given some ordering of these, there is a clear group homomorphism $GL(2,\mathbb{F}_2)\to S_3$.

Since the two groups have the same number of elements, it is enough to show that this map is injective.

Answer 2

Each element of $GL(2,\mathbb F_2)$ can be viewed as a linear transformation $f$ on $\mathbb F_2^2$, which sends $$(1,0)\mapsto x, (0,1)\mapsto y.$$

It follows that $f(1,1)=x+y$ is the only other non-zero vector. Thus $f$ defines a permutation $\sigma_f$ of the three non-zero vectors of $\mathbb F_2^2$.

The mapping $$f\mapsto \sigma_f$$ is clearly a group homomorphism and invertible (take a permutation $\sigma$ on the three non-zero vectors and look at the images of $(1,0)$ and $(0,1)$).