I'm working on the following question from Artin's Algebra:
Let $F=\mathbb{F}_3$. There are four one-dimensional subspaces of the space of column vectors $F^2$.List them. Left multiplication by an invertible matrix permutes these subspaces. Prove that this operation defines a homomorphism $\varphi:GL_2(F) \rightarrow S_4$. Determine the kernel and image of this homomorphism.
Am I on the right track? Here's what I found:
The four one dimensional subspaces are scalars of $(0\;1)^t$, $(1\;0)^t$, $(1\;1)^t$, $(1\;2)^t$, so let $S$ be the set of those 4 vectors.
There exists a bijection between operators of $GL_2(F)$ on $S$ and permutation representations $GL_2(F) \rightarrow S_4$, which is stated in the book. Since we are given such an operation, there exists a homomorphism $\varphi:GL_2(F) \rightarrow S_4$. I count 30 elements in $GL_2(F)$, half of which are scalar multiples of others, so there's a maximum of $15$ elements in the image.
I started a proof by exhaustion of the 15 matrices for each of the four subspaces, but this seems excessive. I'm guessing that the kernel is scalar matrices, so the image is isomorphic to $GL_2(F)/Z_2(F)$
Best Answer
First of all, note that there are $48$ elements in $\operatorname{GL}_2(F)$. After all, we can construct any invertible matrix by first choosing the first column vector, which can be any nonzero vector $v_1\in F$, and then choosing the second column vector, which can be any vector that is linearly independent of $v_1$, i.e. any vector not on the line spanned by $v_1$. This yields a total of $8\times6$ invertible matrices.
Now as you already note, if two matrices in $\operatorname{GL}_2(F)$ differ by a scalar multiple, then their action on the set of lines is the same. What this means is that all scalar matrices are in the kernel of $\varphi$; these matrices act trivially on the set of lines.
Are there any other matrices that act trivially on the set of lines? (Your guess is right). What does this tell you about (the size of) the kernel? And then what does this tell you about (the size of) the image?