Editing Isomorphism theorems

{{Short description|Group of mathematical theorems}}
In [[mathematics]], specifically [[abstract algebra]], the '''isomorphism theorems''' (also known as '''Noether's isomorphism theorems''') are [[theorem]]s that describe the relationship among [[Quotient (universal algebra)|quotients]], [[homomorphism]]s, and [[subobject]]s.  Versions of the theorems exist for [[group (mathematics)|groups]], [[ring (mathematics)|rings]], [[vector space]]s, [[module (mathematics)|modules]], [[Lie algebra]]s, and other [[algebraic structure]]s.  In [[universal algebra]], the isomorphism theorems can be generalized to the context of algebras and [[congruence relation|congruence]]s.

== History ==
The isomorphism theorems were formulated in some generality for homomorphisms of modules by [[Emmy Noether]] in her paper ''Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern'', which was published in 1927 in [[Mathematische Annalen]].  Less general versions of these theorems can be found in work of [[Richard Dedekind]] and previous papers by Noether.

Three years later, [[Bartel Leendert van der Waerden|B.L. van der Waerden]] published his influential ''[[Moderne Algebra]]'', the first [[abstract algebra]] textbook that took the [[Group (mathematics)|groups]]-[[Ring (mathematics)|rings]]-[[Field (mathematics)|fields]] approach to the subject. Van der Waerden credited lectures by Noether on [[group theory]] and [[Emil Artin]] on algebra, as well as a seminar conducted by Artin, [[Wilhelm Blaschke]], [[Otto Schreier]], and van der Waerden himself on [[ideal (ring theory)|ideals]] as the main references. The three isomorphism theorems, called ''homomorphism theorem'', and ''two laws of isomorphism'' when applied to groups, appear explicitly.

== 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.

== Rings ==
The statements of the theorems for [[ring (mathematics)|rings]] are similar, with the notion of a normal subgroup replaced by the notion of an [[ideal (ring theory)|ideal]].

=== Theorem A (rings) ===
Let <math>R</math> and <math>S</math> be rings, and let <math>\varphi:R\rightarrow S</math> be a [[ring homomorphism]]. Then:
# The [[Kernel (algebra)#Ring homomorphisms|kernel]] of <math>\varphi</math> is an ideal of <math>R</math>,
# The [[image (mathematics)|image]] of <math>\varphi</math> is a [[subring]] of <math>S</math>, and
# The image of <math>\varphi</math> is [[ring isomorphism|isomorphic]] to the [[quotient ring]] <math>R/\ker\varphi</math>.
In particular, if <math>\varphi</math> is surjective then <math>S</math> is isomorphic to <math>R/\ker\varphi</math>.<ref>{{Cite web |last=Moy |first=Samuel |date=2022 |title=An Introduction to the Theory of Field Extensions |url=https://math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Moy.pdf |access-date=Dec 20, 2022 |website=UChicago Department of Math}}</ref>

=== Theorem B (rings) ===
Let <math>R</math> be a ring.  Let <math>S</math> be a subring of <math>R</math>, and let <math>I</math> be an ideal of <math>R</math>.  Then:
# The [[Ideal (ring theory)#Ideal operations|sum]] <math>S+I=\{s+i\mid s\in S,i\in I\}</math> is a subring of <math>R</math>,
# The intersection <math>S\cap I</math> is an ideal of <math>S</math>, and
# The quotient rings <math>(S+I)/I</math> and <math>S/(S\cap I)</math> are isomorphic.

=== Theorem C (rings)===
Let ''R'' be a ring, and ''I'' an ideal of ''R''.  Then
# If <math>A</math> is a subring of <math>R</math> such that <math>I \subseteq A \subseteq R</math>, then <math>A/I</math> is a subring of <math>R/I</math>.
# Every subring of <math>R/I</math> is of the form <math>A/I</math> for some subring <math>A</math> of <math>R</math> such that <math>I \subseteq A \subseteq R</math>.
# If <math>J</math> is an ideal of <math>R</math> such that <math>I \subseteq J \subseteq R</math>, then <math>J/I</math> is an ideal of <math>R/I</math>.
# Every ideal of <math>R/I</math> is of the form <math>J/I</math> for some ideal <math>J</math> of <math>R</math> such that <math>I \subseteq J \subseteq R</math>.
# If <math>J</math> is an ideal of <math>R</math> such that <math>I \subseteq J \subseteq R</math>, then the quotient ring <math>(R/I)/(J/I)</math> is isomorphic to <math>R/J</math>.

=== Theorem D (rings)===
Let <math>I</math> be an ideal of <math>R</math>. The correspondence <math>A\leftrightarrow A/I</math> is an [[subset|inclusion]]-preserving [[bijection]] between the set of subrings <math>A</math> of <math>R</math> that contain <math>I</math> and the set of subrings of <math>R/I</math>. Furthermore, <math>A</math> (a subring containing <math>I</math>) is an ideal of <math>R</math> [[if and only if]] <math>A/I</math> is an ideal of <math>R/I</math>.<ref>{{cite book | last1=Dummit | first1=David S. | first2=Richard M. | last2=Foote |  title=Abstract algebra | url=https://archive.org/details/abstractalgebra00dumm_304 | url-access=limited | location=Hoboken, NJ | publisher=Wiley | date=2004 | page=[https://archive.org/details/abstractalgebra00dumm_304/page/n259 246] | isbn=978-0-471-43334-7}}</ref>

== Modules ==
The statements of the isomorphism theorems for [[module (mathematics)|modules]] are particularly simple, since it is possible to form a [[quotient module]] from any [[submodule]].  The isomorphism theorems for [[vector space]]s (modules over a [[field (mathematics)|field]]) and [[abelian group]]s (modules over <math>\mathbb{Z}</math>) are special cases of these.  For [[dimension (vector space)|finite-dimensional]] vector spaces, all of these theorems follow from the [[rank–nullity theorem]].

In the following, "module" will mean "''R''-module" for some fixed ring ''R''.

=== Theorem A (modules) ===
Let <math>M</math> and <math>N</math> be modules, and let <math>\varphi:M\rightarrow N</math> be a [[module homomorphism]].  Then:
# The [[kernel (algebra)|kernel]] of <math>\varphi</math> is a submodule of <math>M</math>,
# The [[image (mathematics)|image]] of <math>\varphi</math> is a submodule of <math>N</math>, and
# The image of <math>\varphi</math> is [[Module_homomorphism#Terminology|isomorphic]] to the [[quotient module]] <math>M/\ker\varphi</math>.
In particular, if <math>\varphi</math> is surjective then <math>N</math> is isomorphic to <math>M/\ker\varphi</math>.

===Theorem B (modules)===
Let <math>M</math> be a module, and let <math>S</math> and <math>T</math> be submodules of <math>M</math>.  Then:
# The sum <math>S+T=\{s+t\mid s\in S,t\in T\}</math> is a submodule of <math>M</math>,
# The intersection <math>S\cap T</math> is a submodule of <math>M</math>, and
# The quotient modules <math>(S+T)/T</math> and <math>S/(S\cap T)</math> are isomorphic.

===Theorem C (modules) ===
Let ''M'' be a module, ''T'' a submodule of ''M''.

# If <math>S</math> is a submodule of <math>M</math> such that <math>T \subseteq S \subseteq M</math>, then <math>S/T</math> is a submodule of <math>M/T</math>.
# Every submodule of <math>M/T</math> is of the form <math>S/T</math> for some submodule <math>S</math> of <math>M</math> such that <math>T \subseteq S \subseteq M</math>.
# If <math>S</math> is a submodule of <math>M</math> such that <math>T \subseteq S \subseteq M</math>, then the quotient module <math>(M/T)/(S/T)</math> is isomorphic to <math>M/S</math>.
<!-- We also need to mention the isomorphism theorems for topological vector spaces, Banach algebras etc. -->

===Theorem D (modules)===
Let <math>M</math> be a module, <math>N</math> a submodule of <math>M</math>. There is a bijection between the submodules of <math>M</math> that contain <math>N</math> and the submodules of <math>M/N</math>. The correspondence is given by <math>A\leftrightarrow A/N</math> for all <math>A\supseteq N</math>. This correspondence commutes with the processes of taking sums and intersections (i.e., is a [[lattice isomorphism]] between the lattice of submodules of <math>M/N</math> and the lattice of submodules of <math>M</math> that contain <math>N</math>).<ref>Dummit and Foote (2004), p. 349</ref>

== Universal algebra ==
To generalise this to [[universal algebra]], normal subgroups need to be replaced by [[congruence relation]]s.

A '''congruence''' on an [[universal algebra|algebra]] <math>A</math> is an [[equivalence relation]] <math>\Phi\subseteq A \times A</math> that forms a subalgebra of <math>A \times A</math> considered as an algebra with componentwise operations.  One can make the set of [[equivalence class]]es <math>A/\Phi</math> into an algebra of the same type by defining the operations via representatives; this will be [[well-defined]] since <math>\Phi</math> is a subalgebra of <math>A \times A</math>. The resulting structure is the [[quotient (universal algebra)|quotient algebra]].

=== Theorem A (universal algebra)===
Let <math>f:A \rightarrow B</math> be an algebra [[homomorphism]].  Then the image of <math>f</math> is a subalgebra of <math>B</math>, the relation given by <math>\Phi:f(x)=f(y)</math> (i.e. the [[Kernel (set theory)|kernel]] of <math>f</math>) is a congruence on <math>A</math>, and the algebras <math>A/\Phi</math> and <math>\operatorname{im} f</math> are [[isomorphic]]. (Note that in the case of a group, <math>f(x)=f(y)</math> [[iff]] <math>f(xy^{-1}) = 1</math>, so one recovers the notion of kernel used in group theory in this case.)

=== Theorem B (universal algebra)===
Given an algebra <math>A</math>, a subalgebra <math>B</math> of <math>A</math>, and a congruence <math>\Phi</math> on <math>A</math>, let <math>\Phi_B = \Phi \cap (B \times B)</math> be the trace of <math>\Phi</math> in <math>B</math> and <math>[B]^\Phi=\{K \in A/\Phi: K \cap B \neq\emptyset\}</math> the collection of equivalence classes that intersect <math>B</math>. Then
# <math>\Phi_B</math> is a congruence on <math>B</math>,
# <math> \ [B]^\Phi</math> is a subalgebra of <math>A/\Phi</math>, and
# the algebra <math>[B]^\Phi</math> is isomorphic to the algebra <math>B/\Phi_B</math>.

=== Theorem C (universal algebra) ===
Let <math>A</math> be an algebra and <math>\Phi, \Psi</math> two congruence relations on <math>A</math> such that <math>\Psi \subseteq \Phi</math>. Then <math>\Phi/\Psi = \{ ([a']_\Psi,[a'']_\Psi): (a',a'')\in \Phi\} = [\ ]_\Psi \circ \Phi \circ [\ ]_\Psi^{-1}</math> is a congruence on <math>A/\Psi</math>, and <math>A/\Phi</math> is isomorphic to <math>(A/\Psi)/(\Phi/\Psi).</math>

=== Theorem D (universal algebra) ===
Let <math>A</math> be an algebra and denote <math>\operatorname{Con}A</math> the set of all congruences on <math>A</math>. The set
<math>\operatorname{Con}A</math> is a [[complete lattice]] ordered by inclusion.<ref>Burris and Sankappanavar (2012), p. 37</ref>
If <math>\Phi\in\operatorname{Con}A</math> is a congruence and we denote by <math>\left[\Phi,A\times A\right]\subseteq\operatorname{Con}A</math> the set of all congruences that contain <math>\Phi</math> (i.e. <math>\left[\Phi,A\times A\right]</math> is a principal [[filter (mathematics)|filter]] in <math>\operatorname{Con}A</math>, moreover it is a sublattice), then
the map  <math>\alpha:\left[\Phi,A\times A\right]\to\operatorname{Con}(A/\Phi),\Psi\mapsto\Psi/\Phi</math> is a lattice isomorphism.<ref>Burris and Sankappanavar (2012), p. 49</ref><ref>{{cite web |first1=William |last1=Sun |title=Is there a general form of the correspondence theorem? |url=https://math.stackexchange.com/q/2850331 |website=Mathematics StackExchange |access-date=20 July 2019}}</ref>

== Notes ==
{{Reflist}}

== References ==
* [[Noether, Emmy]], ''Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern'', [[Mathematische Annalen]] '''96'''  (1927) pp.&nbsp;26–61
* [[McLarty, Colin]], "Emmy Noether's 'Set Theoretic' Topology: From Dedekind to the rise of functors". ''The Architecture of Modern Mathematics: Essays in history and philosophy'' (edited by [[Jeremy Gray]] and José Ferreirós), Oxford University Press (2006) pp.&nbsp;211–35.
* {{Citation| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| date=2009| title=Basic algebra| edition= 2nd| volume = 1 | publisher=Dover| isbn = 9780486471891}}
* Cohn, Paul M., ''Universal algebra'', Chapter II.3 p.&nbsp;57
* {{Citation | last=Milne | first=James S. | title=Group Theory | year=2013 | version = 3.13 |  url=http://www.jmilne.org/math/}}
* {{citation | last=van der Waerden | first=B. I. | title=[[Moderne Algebra|Algebra]] | edition= 9 | volume=1 | publisher=Springer-Verlag | year=1994}}
* {{cite book | last1=Dummit | first1=David S. | first2=Richard M. | last2=Foote |  title=Abstract algebra | location=Hoboken, NJ | publisher=Wiley | date=2004 | isbn=978-0-471-43334-7}}
* {{cite book |last1=Burris |first1=Stanley |last2=Sankappanavar |first2=H. P. |title=A Course in Universal Algebra |date=2012 |url=https://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf |isbn=978-0-9880552-0-9}}
* {{citation | author=Scott, W. R. | title=Group Theory | publisher=Prentice Hall | year=1964}}
* {{cite book|author=Durbin, John R.|title=Modern Algebra: An Introduction|year=2009|edition= 6|publisher=Wiley|isbn=978-0-470-38443-5}}
* {{citation | title=Basic Algebra | author= Knapp, Anthony W. | edition= Digital second | year=2016}}
* {{citation | author=Grillet, Pierre Antoine | title=Abstract Algebra| edition= 2 | publisher=Springer | year=2007}}
* {{citation | author = Rotman, Joseph J. | title=Advanced Modern Algebra | publisher=Prentice Hall | edition= 2 | year=2003 | isbn=0130878685 }}
* {{citation | author = Hungerford, Thomas W. | title=Algebra (Graduate Texts in Mathematics, 73) | publisher=Springer | year=1980 | isbn=0387905189 }}

[[Category:Isomorphism theorems| ]]