Editing Isomorphism theorems (section)

== Groups ==
We first present the isomorphism theorems of the [[group (mathematics)|groups]].

===Theorem A (groups)===
{{see also | Fundamental theorem on homomorphisms}}

[[Image:First-isomorphism-theorem.svg|thumb|Diagram of the fundamental theorem on homomorphisms]]

Let <math>G</math> and <math>H</math> be groups, and let <math> f : G \rightarrow H</math> be a [[group homomorphism|homomorphism]].  Then:
# The [[Kernel (algebra)#Group homomorphisms|kernel]] of <math>f</math> is a [[normal subgroup]] of <math>G</math>,
# The [[image (mathematics)|image]] of <math>f</math> is a [[subgroup]] of <math>H</math>, and
# The image of <math>f</math> is [[group isomorphism|isomorphic]] to the [[quotient group]] <math>G / \ker f</math>.
In particular, if <math>f</math> is [[surjective]] then <math>H</math> is isomorphic to <math>G / \ker f</math>.

This theorem is usually called the {{anchor|first isomorphism theorem}}''first isomorphism theorem''.

===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>

===Theorem C (groups)===
Let <math>G</math> be a group, and <math>N</math> a normal subgroup of <math>G</math>.
Then
# If <math>K</math> is a subgroup of <math>G</math> such that <math>N \subseteq K \subseteq G</math>, then <math>G/N</math> has a subgroup isomorphic to <math>K/N</math>.
# Every subgroup of <math>G/N</math> is of the form <math>K/N</math> for some subgroup <math>K</math> of <math>G</math> such that <math>N \subseteq K \subseteq G</math>.
# If <math>K</math> is a normal subgroup of <math>G</math> such that <math>N \subseteq K \subseteq G</math>, then <math>G/N</math> has a normal subgroup isomorphic to <math>K/N</math>.
# Every normal subgroup of <math>G/N</math> is of the form <math>K/N</math> for some normal subgroup <math>K</math> of <math>G</math> such that <math>N \subseteq K \subseteq G</math>.
# If <math>K</math> is a normal subgroup of <math>G</math> such that <math>N \subseteq K \subseteq G</math>, then the quotient group <math>(G/N)/(K/N)</math> is isomorphic to <math>G/K</math>.

The last statement is sometimes referred to as the ''third isomorphism theorem''. The first four statements are often subsumed under Theorem D below, and referred to as the ''lattice theorem'', ''correspondence theorem'', or ''fourth isomorphism theorem''.

===Theorem D (groups)===

{{main | Lattice theorem}}
Let <math>G</math> be a group, and <math>N</math> a normal subgroup of <math>G</math>.
The canonical projection homomorphism <math>G\rightarrow G/N</math> defines a bijective correspondence
between the set of subgroups of <math>G</math> containing <math>N</math> and the set of (all) subgroups of <math>G/N</math>. Under this correspondence normal subgroups correspond to normal subgroups. 

This theorem is sometimes called the [[Correspondence theorem (group theory)|''correspondence theorem'']], the ''lattice theorem'', and the ''fourth isomorphism theorem''.

The [[Zassenhaus lemma]] (also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem.<ref>{{cite book |last=Wilson |first=Robert A. |author-link=Robert A. Wilson (mathematician) |title=The Finite Simple Groups |date=2009 |doi=10.1007/978-1-84800-988-2 |at=p. 7 |publisher=Springer-Verlag London |isbn=978-1-4471-2527-3 |series=Graduate Texts in Mathematics 251|volume=251 }}</ref>

=== Discussion ===

The first isomorphism theorem can be expressed in [[category theory|category theoretical]] language by saying that the [[category of groups]] is (normal epi, mono)-factorizable; in other words, the [[normal morphism|normal epimorphisms]] and the [[monomorphism]]s form a [[factorization system]] for the [[category (mathematics)|category]].  This is captured in the [[commutative diagram]] in the margin, which shows the [[object (category theory)|objects]] and [[morphism]]s whose existence can be deduced from the morphism <math> f : G \rightarrow H</math>.  The diagram shows that every morphism in the category of groups has a [[Kernel (category theory)|kernel]] in the category theoretical sense; the arbitrary morphism ''f'' factors into <math>\iota \circ \pi</math>, where ''ι'' is a monomorphism and ''π'' is an epimorphism (in a [[conormal category]], all epimorphisms are normal).  This is represented in the diagram by an object <math>\ker f</math> and a monomorphism <math>\kappa: \ker f \rightarrow G</math> (kernels are always monomorphisms), which complete the [[short exact sequence]] running from the lower left to the upper right of the diagram.  The use of the [[exact sequence]] convention saves us from having to draw the [[zero morphism]]s from <math>\ker f</math> to <math>H</math> and <math>G / \ker f</math>.

If the sequence is right split (i.e., there is a morphism ''σ'' that maps <math>G / \operatorname{ker} f</math> to a {{pi}}-preimage of itself), then ''G'' is the [[semidirect product]] of the normal subgroup <math>\operatorname{im} \kappa</math> and the subgroup <math>\operatorname{im} \sigma</math>.  If it is left split (i.e., there exists some <math>\rho: G \rightarrow \operatorname{ker} f</math> such that <math>\rho \circ \kappa = \operatorname{id}_{\text{ker} f}</math>), then it must also be right split, and <math>\operatorname{im} \kappa \times \operatorname{im} \sigma</math> is a [[direct product]] decomposition of ''G''.  In general, the existence of a right split does not imply the existence of a left split; but in an [[abelian category]] (such as [[category of abelian groups|that of abelian groups]]), left splits and right splits are equivalent by the [[splitting lemma]], and a right split is sufficient to produce a [[Direct sum of groups|direct sum]] decomposition <math>\operatorname{im} \kappa \oplus \operatorname{im} \sigma</math>.  In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence <math>0 \rightarrow G / \operatorname{ker} f \rightarrow H \rightarrow \operatorname{coker} f \rightarrow 0</math>.

In the second isomorphism theorem, the product ''SN'' is the [[join and meet|join]] of ''S'' and ''N'' in the [[lattice of subgroups]] of ''G'', while the intersection ''S''&nbsp;∩&nbsp;''N'' is the [[join and meet|meet]].

The third isomorphism theorem is generalized by the [[nine lemma]] to [[abelian categories]] and more general maps between objects.

=== Note on numbers and names ===
Below we present four theorems, labelled A, B, C and D. They are often numbered as "First isomorphism theorem", "Second..." and so on; however, there is no universal agreement on the numbering. Here we give some examples of the group isomorphism theorems in the literature. Notice that these theorems have analogs for rings and modules.

<!-- Do not expand this list indiscriminately. This is here just to show the lack of the established convention. -->
{| class="wikitable"
|+ Comparison of the names of the group isomorphism theorems
|-
! scope="col" | Comment
! scope="col" | Author
! scope="col" | Theorem A
! scope="col" | Theorem B
! scope="col" | Theorem C
|-
| rowspan=4 | No "third" theorem
! Jacobson<ref>Jacobson (2009), sec 1.10</ref>
| Fundamental theorem of homomorphisms
| (''Second isomorphism theorem'')
| "often called the first isomorphism theorem"
|-
! van der Waerden,<ref>van der Waerden, ''[[Moderne Algebra|Algebra]]'' (1994).</ref> Durbin{{refn| ''[the names are] essentially the same as [van der Waerden 1994]''<ref>Durbin (2009), sec. 54</ref>}}
| Fundamental theorem of homomorphisms
| First isomorphism theorem
| Second isomorphism theorem
|-
! Knapp<ref>Knapp (2016), sec IV 2</ref>
| (''No name'')
| Second isomorphism theorem
| First isomorphism theorem
|-
! Grillet<ref>Grillet (2007), sec. I 5</ref>
| Homomorphism theorem
| Second isomorphism theorem
| First isomorphism theorem
|-
| rowspan=4 | Three numbered theorems
! (''Other convention per Grillet'')
| First isomorphism theorem
| Third isomorphism theorem
| Second isomorphism theorem
|-
! Rotman<ref>Rotman (2003), sec. 2.6</ref>
| First isomorphism theorem
| Second isomorphism theorem
| Third isomorphism theorem
|-
! Fraleigh<ref>Fraleigh (2003), Chap. 14, 34</ref>
| Fundamental homomorphism theorem or first isomorphism theorem
| Second isomorphism theorem
| Third isomorphism theorem
|-
! Dummit & Foote<ref>{{Cite book|last=Dummit|first=David Steven|url=https://www.worldcat.org/oclc/52559229|title=Abstract algebra|date=2004|others=Richard M. Foote|isbn=0-471-43334-9|edition=Third|publisher=John Wiley and Sons, Inc.|location=Hoboken, NJ|pages=97–98|oclc=52559229}}</ref>
| First isomorphism theorem
| Second or Diamond isomorphism theorem
| Third isomorphism theorem
|-
| rowspan=2 | No numbering
! Milne<ref name="milne">Milne (2013), Chap. 1, sec. ''Theorems concerning homomorphisms''</ref>
| Homomorphism theorem
| Isomorphism theorem
| Correspondence theorem
|-
! Scott<ref>Scott (1964), secs 2.2 and 2.3</ref>
| Homomorphism theorem
| Isomorphism theorem
| Freshman theorem
|}
<!-- Do not expand this list indiscriminantly. This is here just to show the lack of the established convention. -->
It is less common to include the Theorem D, usually known as the ''[[lattice theorem]]'' or the ''correspondence theorem'', as one of isomorphism theorems, but when included, it is the last one.