= SL2 over F2 as a permutation group
{c}
{title2=$\mathrm{SL}_2(\mathbb F_2)\cong S_3$}
A determinant-one matrix over $\mathbb F_2$ permutes the three nonzero vectors of $\mathbb F_2^2$. This <group action> is faithful because fixing two basis vectors forces the matrix to be the identity. There are $(4-1)(4-2)=6$ invertible matrices, all of determinant one since the only nonzero field element is one, so the action identifies the group with the <symmetric group> $S_3$. Reduction of integer determinant-one matrices modulo two is onto: $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ and $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ induce two distinct transpositions, which generate $S_3$. Its kernel consists of matrices congruent to the identity modulo two, the level-two <principal congruence subgroup>.
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