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.
Best Answer
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.