= SL2 action on a finite projective line
{c}
{title2=$\mathrm{SL}_2(\mathbb F_q)\curvearrowright\mathbb P^1(\mathbb F_q)$}
A matrix $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ acts on the <projective line> by the <Möbius transformation> $x\mapsto(ax+b)/(cx+d)$, with the pole sent to infinity and infinity sent to $a/c$ when $c\ne0$. The action is transitive because $\begin{pmatrix}t&-1\\1&0\end{pmatrix}$ sends infinity to any finite $t$. The <stabilizer subgroup> of infinity consists of $\begin{pmatrix}a&b\\0&a^{-1}\end{pmatrix}$ with $a\ne0$. The <orbit-stabilizer theorem> gives $|\mathrm{SL}_2(\mathbb F_q)|=q(q^2-1)$. Its kernel is the scalar matrices of determinant one: a transformation fixing infinity has $c=0$, fixing zero then forces $b=0$, and fixing one gives $a=d$. Thus the kernel is $\{I,-I\}$, with these matrices coinciding in characteristic two.
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