information:extraction
Theorem B (groups)

Let G {\displaystyle G} be a group. Let S {\displaystyle S} be a subgroup of G {\displaystyle G}, and let N {\displaystyle N} be a normal subgroup of G {\displaystyle G}. Then the following hold:

Technically, it is not necessary for N {\displaystyle N} to be a normal subgroup, as long as S {\displaystyle S} is a subgroup of the normalizer of N {\displaystyle N} in G {\displaystyle G}. In this case, N {\displaystyle N} is not a normal subgroup of G {\displaystyle G}, but N {\displaystyle N} is still a normal subgroup of the product S N {\displaystyle SN}.

This theorem is sometimes called the second isomorphism theorem, diamond theorem or the parallelogram theorem.

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