Editing Isomorphism theorems (section)

===Theorem B (groups)===
[[File:Diagram for the First Isomorphism Theorem.png|thumb|Diagram for theorem B4. The two quotient groups (dotted) are isomorphic.]]
Let <math>G</math> be a group.  Let <math>S</math> be a subgroup of <math>G</math>, and let <math>N</math> be a normal subgroup of <math>G</math>.  Then the following hold:
# The [[product of group subsets|product]] <math>SN</math> is a subgroup of <math>G</math>,
# The subgroup <math>N</math> is a normal subgroup of <math>SN</math>,
# The [[intersection (set theory)|intersection]] <math>S \cap N</math> is a normal subgroup of <math>S</math>, and
# The quotient groups <math>(SN)/N</math> and <math>S/(S\cap N)</math> are isomorphic.
Technically, it is not necessary for <math>N</math> to be a normal subgroup, as long as <math>S</math> is a subgroup of the [[normalizer]] of <math>N</math> in <math>G</math>.  In this case, <math>N</math> is not a normal subgroup of <math>G</math>, but <math>N</math> is still a normal subgroup of the product <math>SN</math>.

This theorem is sometimes called the ''second isomorphism theorem'',<ref name="milne"/> ''diamond theorem''<ref name="Isaacs1994">{{cite book|author=I. Martin Isaacs|author-link=Martin Isaacs|title=Algebra: A Graduate Course|url=https://archive.org/details/algebragraduatec00isaa|url-access=limited|year=1994|publisher=American Mathematical Soc.|isbn=978-0-8218-4799-2|page=[https://archive.org/details/algebragraduatec00isaa/page/n45 33]}}</ref> or the ''parallelogram theorem''.<ref name="Cohn2000">{{cite book|author=Paul Moritz Cohn|author-link=Paul Moritz Cohn|title=Classic Algebra|url=https://archive.org/details/classicalgebra00cohn_300|url-access=limited|year=2000|publisher=Wiley|isbn=978-0-471-87731-8|page=[https://archive.org/details/classicalgebra00cohn_300/page/n256 245]}}</ref>

An application of the second isomorphism theorem identifies [[projective linear group]]s: for example, the group on the [[complex projective line]] starts with setting <math>G = \operatorname{GL}_2(\mathbb{C})</math>, the group of [[invertible matrix|invertible]] 2&thinsp;×&thinsp;2 [[complex number|complex]] [[matrix (mathematics)|matrices]], <math>S = \operatorname{SL}_2(\mathbb{C})</math>, the subgroup of [[determinant]] 1 matrices, and <math>N</math> the normal subgroup of scalar matrices <math>\mathbb{C}^{\times}\!I = \left\{\left( \begin{smallmatrix} a & 0 \\ 0 & a \end{smallmatrix} \right) : a \in \mathbb{C}^{\times} \right\}</math>, we have <math>S \cap N = \{\pm I\}</math>, where <math>I</math> is the [[identity matrix]], and <math>SN = \operatorname{GL}_2(\mathbb{C})</math>. Then the second isomorphism theorem states that:

: <math>\operatorname{PGL}_2(\mathbb{C}) := \operatorname{GL}_2 \left(\mathbb{C})/(\mathbb{C}^{\times}\!I\right) \cong \operatorname{SL}_2(\mathbb{C})/\{\pm I\} =: \operatorname{PSL}_2(\mathbb{C})</math>