Second isomorphism theorem

Theorem B (groups)

An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting ${\displaystyle G=\operatorname {GL} _{2}(\mathbb {C} )}$, the group of invertible 2 × 2 complexmatrices, ${\displaystyle S=\operatorname {SL} _{2}(\mathbb {C} )}$, the subgroup of determinant 1 matrices, and ${\displaystyle N}$ the normal subgroup of scalar matrices ${\displaystyle \mathbb {C} ^{\times }\!I=\left\{\left({\begin{smallmatrix}a&0\\0&a\end{smallmatrix}}\right):a\in \mathbb {C} ^{\times }\right\}}$, we have ${\displaystyle S\cap N=\{\pm I\}}$, where ${\displaystyle I}$ is the identity matrix, and ${\displaystyle SN=\operatorname {GL} _{2}(\mathbb {C} )}$. Then the second isomorphism theorem states that: