Editing Normal subgroup

{{Short description|Subgroup invariant under conjugation}}
{{Redirect-distinguish|Invariant subgroup|Fully invariant subgroup}}
{{Group theory sidebar|Basics}}
In [[abstract algebra]], a '''normal subgroup''' (also known as an '''invariant subgroup''' or '''self-conjugate subgroup'''){{sfn|Bradley|2010|p=12}} is a [[subgroup]] that is [[Invariant (mathematics)|invariant]] under [[inner automorphism|conjugation]] by members of the [[Group (mathematics)|group]] of which it is a part. In other words, a subgroup <math>N</math> of the group <math>G</math> is normal in <math>G</math> if and only if <math>gng^{-1} \in N</math> for all <math>g \in G</math> and <math>n \in N.</math> The usual notation for this relation is <math>N \triangleleft G.</math>

Normal subgroups are important because they (and only they) can be used to construct [[quotient group]]s of the given group.  Furthermore, the normal subgroups of <math>G</math> are precisely the [[Kernel of a homomorphism|kernels]] of [[Group homomorphism|group homomorphisms]] with [[domain of a function|domain]] <math>G,</math> which means that they can be used to internally classify those homomorphisms.

[[Évariste Galois]] was the first to realize the importance of the existence of normal subgroups.{{sfn|Cantrell|2000|p=160}}

== Definitions ==
A [[subgroup]] <math>N</math> of a group <math>G</math> is called a '''normal subgroup''' of <math>G</math> if it is invariant under [[inner automorphism|conjugation]]; that is, the conjugation of an element of <math>N</math> by an element of <math>G</math> is always in <math>N.</math>{{sfn|Dummit|Foote|2004}}  The usual notation for this relation is <math>N \triangleleft G.</math>

===Equivalent conditions===
For any subgroup <math>N</math> of <math>G,</math> the following conditions are [[Logical equivalence|equivalent]] to <math>N</math> being a normal subgroup of <math>G.</math> Therefore, any one of them may be taken as the definition.

* The image of conjugation of <math>N</math> by any element of <math>G</math> is a subset of <math>N,</math>{{sfn|Hungerford|2003|p=41}} i.e., <math>gNg^{-1}\subseteq N</math> for all <math>g\in G</math>.
* The image of conjugation of <math>N</math> by any element of <math>G</math> is equal to <math>N,</math>{{sfn|Hungerford|2003|p=41}} i.e., <math>gNg^{-1}= N</math> for all <math>g\in G</math>.
* For all <math>g \in G,</math> the left and right cosets <math>gN</math> and <math>Ng</math> are equal.{{sfn|Hungerford|2003|p=41}}
* The sets of left and right [[coset]]s of <math>N</math> in <math>G</math> coincide.{{sfn|Hungerford|2003|p=41}}
* Multiplication in <math>G</math> preserves the equivalence relation "is in the same left coset as". That is, for every <math>g,g',h,h'\in G</math> satisfying <math>g N = g' N</math> and <math>h N = h' N</math>, we have <math>(g h) N = (g' h') N.</math>
* There exists a group on the set of left cosets of <math>N</math> where multiplication of any two left cosets <math>gN</math> and <math>hN</math> yields the left coset <math>(gh)N</math>. (This group is called the ''quotient group'' of <math>G</math> ''modulo'' <math>N</math>, denoted <math>G/N</math>.)
* <math>N</math> is a [[Union (set theory)|union]] of [[conjugacy class]]es of <math>G.</math>{{sfn|Cantrell|2000|p=160}}
* <math>N</math> is preserved by the [[inner automorphism]]s of <math>G.</math>{{sfn|Fraleigh|2003|p=141}}
* There is some [[group homomorphism]] <math>G \to H</math> whose [[Kernel (algebra)|kernel]] is <math>N.</math>{{sfn|Cantrell|2000|p=160}}
* There exists a group homomorphism <math>\phi:G \to H</math> whose [[Fiber (mathematics)|fibers]] form a group where the identity element is <math>N</math> and multiplication of any two fibers <math>\phi^{-1}(h_1)</math> and <math>\phi^{-1}(h_2)</math> yields the fiber <math>\phi^{-1}(h_1 h_2)</math>. (This group is the same group <math>G/N</math> mentioned above.)
* There is some [[congruence relation]] on <math>G</math> for which the [[equivalence class]] of the [[identity element]] is <math>N</math>.
* For all <math>n\in N</math> and <math>g\in G,</math> the [[commutator]] <math>[n,g] = n^{-1} g^{-1} n g</math> is in <math>N.</math>{{cn|date=March 2019}}
* Any two elements commute modulo the normal subgroup membership relation.  That is, for all <math>g, h \in G,</math> <math>g h \in N</math> if and only if <math>h g \in N.</math>{{cn|date=October 2020}}

== Examples ==
For any group <math>G,</math> the trivial subgroup <math>\{ e \}</math> consisting of just the identity element of <math>G</math> is always a normal subgroup of <math>G.</math>  Likewise, <math>G</math> itself is always a normal subgroup of <math>G.</math> (If these are the only normal subgroups, then <math>G</math> is said to be [[Simple group|simple]].){{sfn|Robinson|1996|p=16}}  Other named normal subgroups of an arbitrary group include the [[Center (group theory)|center of the group]] (the set of elements that commute with all other elements) and the [[commutator subgroup]] <math>[G,G].</math>{{sfn|Hungerford|2003|p=45}}{{sfn|Hall|1999|p=138}}  More generally, since conjugation is an isomorphism, any [[characteristic subgroup]] is a normal subgroup.{{sfn|Hall|1999|p=32}}

If <math>G</math> is an [[abelian group]] then every subgroup <math>N</math> of <math>G</math> is normal, because <math>gN = \{gn\}_{n\in N} = \{ng\}_{n\in N} = Ng.</math> More generally, for any group <math>G</math>, every subgroup of the ''center'' <math>Z(G)</math> of <math>G</math> is normal in <math>G</math>. (In the special case that <math>G</math> is abelian, the center is all of <math>G</math>, hence the fact that all subgroups of an abelian group are normal.)  A group that is not abelian but for which every subgroup is normal is called a [[Hamiltonian group]].{{sfn|Hall|1999|p=190}}

A concrete example of a normal subgroup is the subgroup <math>N = \{(1), (123), (132)\}</math> of the [[symmetric group]] <math>S_3,</math> consisting of the identity and both three-cycles.  In particular, one can check that every coset of <math>N</math> is either equal to <math>N</math> itself or is equal to <math>(12)N = \{ (12), (23), (13)\}.</math>  On the other hand, the subgroup <math>H = \{(1), (12)\}</math> is not normal in <math>S_3</math> since <math>(123)H = \{(123), (13) \} \neq \{(123), (23) \} = H(123).</math>{{sfn|Judson|2020|loc = Section 10.1}} This illustrates the general fact that any subgroup <math>H \leq G</math> of index two is normal.

As an example of a normal subgroup within a [[matrix group]], consider the [[general linear group]] <math>\mathrm{GL}_n(\mathbf{R})</math> of all invertible <math>n\times n</math> matrices with real entries under the operation of matrix multiplication and its subgroup <math>\mathrm{SL}_n(\mathbf{R})</math> of all <math>n\times n</math> matrices of [[determinant (mathematics)|determinant]] 1 (the [[special linear group]]).  To see why the subgroup <math>\mathrm{SL}_n(\mathbf{R})</math> is normal in <math>\mathrm{GL}_n(\mathbf{R})</math>, consider any matrix <math>X</math> in <math>\mathrm{SL}_n(\mathbf{R})</math> and any invertible matrix <math>A</math>.  Then using the two important identities <math>\det(AB)=\det(A)\det(B)</math> and <math>\det(A^{-1})=\det(A)^{-1}</math>, one has that <math>\det(AXA^{-1}) = \det(A) \det(X) \det(A)^{-1} = \det(X) = 1</math>, and so <math>AXA^{-1} \in \mathrm{SL}_n(\mathbf{R})</math> as well.  This means  <math>\mathrm{SL}_n(\mathbf{R})</math> is closed under conjugation in <math>\mathrm{GL}_n(\mathbf{R})</math>, so it is a normal subgroup.{{efn|In other language: <math>\det</math> is a homomorphism from <math>\mathrm{GL}_n(\mathbf{R})</math> to the multiplicative subgroup <math> \mathbf{R}^\times</math>, and <math>\mathrm{SL}_n(\mathbf{R})</math> is the kernel.  Both arguments also work over the [[complex number]]s, or indeed over an arbitrary [[field (mathematics)|field]].}}

In the [[Rubik's Cube group]], the subgroups consisting of operations which only affect the orientations of either the corner pieces or the edge pieces are normal.{{sfn|Bergvall|Hynning|Hedberg|Mickelin|2010|p=96}}

The [[translation group]] is a normal subgroup of the [[Euclidean group]] in any dimension.{{sfn|Thurston|1997|p=218}} This means: applying a rigid transformation, followed by a translation and then the inverse rigid transformation, has the same effect as a single translation. By contrast, the subgroup of all [[Rotation|rotations]] about the origin is ''not'' a normal subgroup of the Euclidean group, as long as the dimension is at least 2: first translating, then rotating about the origin, and then translating back will typically not fix the origin and will therefore not have the same effect as a single rotation about the origin.

== Properties ==

* If <math>H</math> is a normal subgroup of <math>G,</math> and <math>K</math> is a subgroup of <math>G</math> containing <math>H,</math> then <math>H</math> is a normal subgroup of <math>K.</math>{{sfn|Hungerford|2003|p=42}}
* A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a [[transitive relation]]. The smallest group exhibiting this phenomenon is the [[dihedral group]] of order 8.{{sfn|Robinson|1996|p=17}} However, a [[characteristic subgroup]] of a normal subgroup is normal.{{sfn|Robinson|1996|p=28}} A group in which normality is transitive is called a [[T-group (mathematics)|T-group]].{{sfn|Robinson|1996|p=402}}
* The two groups <math>G</math> and <math>H</math> are normal subgroups of their [[Direct product of groups|direct product]] <math>G \times H.</math>
* If the group <math>G</math> is a [[semidirect product]] <math>G = N \rtimes H,</math> then <math>N</math> is normal in <math>G,</math> though <math>H</math> need not be normal in <math>G.</math>
* If <math>M</math> and <math>N</math> are normal subgroups of an additive group <math>G</math> such that <math>G = M + N</math> and <math>M \cap N = \{0\}</math>, then <math>G = M \oplus N.</math>{{sfn|Hungerford|2013|p=290}}
* Normality is preserved under surjective homomorphisms;{{sfn|Hall|1999|p=29}} that is, if <math>G \to H</math> is a surjective group homomorphism and <math>N</math> is normal in <math>G,</math> then the image <math>f(N)</math> is normal in <math>H.</math>
* Normality is preserved by taking [[Inverse image|inverse images]];{{sfn|Hall|1999|p=29}} that is, if <math>G \to H</math> is a group homomorphism and <math>N</math> is normal in <math>H,</math> then the inverse image <math>f^{-1}(N)</math> is normal in <math>G.</math>
* Normality is preserved on taking [[direct product of groups|direct products]];{{sfn|Hungerford|2003|p=46}} that is, if <math>N_1 \triangleleft G_1</math> and <math>N_2 \triangleleft G_2,</math> then <math>N_1 \times N_2\; \triangleleft \;G_1 \times G_2.</math>
* Every subgroup of [[Index (group theory)|index]] 2 is normal. More generally, a subgroup, <math>H,</math> of finite index, <math>n,</math> in <math>G</math> contains a subgroup, <math>K,</math> normal in <math>G</math> and of index dividing <math>n!</math> called the [[normal core]]. In particular, if <math>p</math> is the smallest prime dividing the order of <math>G,</math> then every subgroup of index <math>p</math> is normal.{{sfn|Robinson|1996|p=36}}
* The fact that normal subgroups of <math>G</math> are precisely the kernels of group homomorphisms defined on <math>G</math> accounts for some of the importance of normal subgroups; they are a way to internally classify all homomorphisms defined on a group. For example, a non-identity finite group is [[Simple group|simple]] if and only if it is isomorphic to all of its non-identity homomorphic images,{{sfn|Dõmõsi|Nehaniv|2004|p=7}} a finite group is [[Perfect group|perfect]] if and only if it has no normal subgroups of prime [[Index of a subgroup|index]], and a group is [[Imperfect group|imperfect]] if and only if the [[derived subgroup]] is not supplemented by any proper normal subgroup.

=== Lattice of normal subgroups ===
Given two normal subgroups, <math>N</math> and <math>M,</math> of <math>G,</math> their intersection <math>N\cap M</math>and their [[Product of subgroups|product]] <math>N M = \{n m : n \in N\; \text{ and }\; m \in M \}</math> are also normal subgroups of <math>G.</math> 

The normal subgroups of <math>G</math> form a [[Lattice (order)|lattice]] under [[subset inclusion]] with [[least element]], <math>\{ e \},</math> and [[greatest element]], <math>G.</math> The [[Meet (lattice theory)|meet]] of two normal subgroups, <math>N</math> and <math>M,</math> in this lattice is their intersection and the [[Join (lattice theory)|join]] is their product.

The lattice is [[Complete lattice|complete]] and [[Modular lattice|modular]].{{sfn|Hungerford|2003|p=46}}

== Normal subgroups, quotient groups and homomorphisms ==

If <math>N</math> is a normal subgroup, we can define a multiplication on cosets as follows: 
<math display="block">\left(a_1 N\right) \left(a_2 N\right) := \left(a_1 a_2\right) N.</math>
This relation defines a mapping <math>G/N\times G/N \to G/N.</math> To show that this mapping is well-defined, one needs to prove that the choice of representative elements <math>a_1, a_2</math> does not affect the result. To this end, consider some other representative elements <math>a_1'\in a_1 N, a_2' \in a_2 N.</math> Then there are <math>n_1, n_2\in N</math> such that <math>a_1' = a_1 n_1, a_2' = a_2 n_2.</math> It follows that <math display="block">a_1' a_2' N = a_1 n_1 a_2 n_2 N =a_1 a_2 n_1' n_2 N=a_1 a_2 N,</math>where we also used the fact that <math>N</math> is a {{em|normal}} subgroup, and therefore there is <math>n_1'\in N</math> such that <math>n_1 a_2 = a_2 n_1'.</math> This proves that this product is a well-defined mapping between cosets.

With this operation, the set of cosets is itself a group, called the [[quotient group]] and denoted with <math>G/N.</math> There is a natural [[Group homomorphism|homomorphism]], <math>f : G \to G/N,</math> given by <math>f(a) = a N.</math> This homomorphism maps <math>N</math> into the identity element of <math>G/N,</math> which is the coset <math>e N = N,</math>{{sfn|Hungerford|2003|pp=42–43}} that is, <math>\ker(f) = N.</math>

In general, a group homomorphism, <math>f : G \to H</math> sends subgroups of <math>G</math> to subgroups of <math>H.</math> Also, the preimage of any subgroup of <math>H</math> is a subgroup of <math>G.</math> We call the preimage of the trivial group <math>\{ e \}</math> in <math>H</math> the '''[[Kernel (algebra)|kernel]]''' of the homomorphism and denote it by <math>\ker f.</math> As it turns out, the kernel is always normal and the image of <math>G, f(G),</math> is always [[isomorphic]] to <math>G / \ker f</math> (the [[first isomorphism theorem]]).{{sfn|Hungerford|2003|p=44}} In fact, this correspondence is a bijection between the set of all quotient groups of <math>G, G / N,</math> and the set of all homomorphic images of <math>G</math> ([[up to]] isomorphism).{{sfn|Robinson|1996|p=20}} It is also easy to see that the kernel of the quotient map, <math>f : G \to G/N,</math> is <math>N</math> itself, so the normal subgroups are precisely the kernels of homomorphisms with [[Domain of a function|domain]] <math>G.</math>{{sfn|Hall|1999|p=27}}

==See also==
{{div col}}

===Operations taking subgroups to subgroups===
*[[Normalizer]]
*[[Normal closure (group theory)|Normal closure]]
*[[Normal core]]

===Subgroup properties complementary (or opposite) to normality===
*[[Malnormal subgroup]]
*[[Contranormal subgroup]]
*[[Abnormal subgroup]]
*[[Self-normalizing subgroup]]

===Subgroup properties stronger than normality===
*[[Characteristic subgroup]]
*[[Fully characteristic subgroup]]

===Subgroup properties weaker than normality===
*[[Subnormal subgroup]]
*[[Ascendant subgroup]]
*[[Descendant subgroup]]
*[[Quasinormal subgroup]]
*[[Seminormal subgroup]]
*[[Conjugate permutable subgroup]]
*[[Modular subgroup]]
*[[Pronormal subgroup]]
*[[Paranormal subgroup]]
*[[Polynormal subgroup]]
*[[C-normal subgroup]]

===Related notions in algebra===
*[[Ideal (ring theory)]]
*[[Semigroup ideal]]

{{div col end}}

== Notes ==
{{notelist}}
== References ==
{{reflist}}

== Bibliography ==
{{refbegin|30em}}
*{{cite web|last1=Bergvall|first1=Olof|last2=Hynning|first2=Elin|last3=Hedberg|first3=Mikael|last4=Mickelin|first4=Joel|last5=Masawe|first5=Patrick|title=On Rubik's Cube|date=16 May 2010|publisher=[[KTH Royal Institute of Technology|KTH]]|url=https://people.kth.se/~boij/kandexjobbVT11/Material/rubikscube.pdf}}
*{{cite book|last=Cantrell|first=C.D.|title=Modern Mathematical Methods for Physicists and Engineers|url=https://archive.org/details/modernmathematic0000cant|url-access=registration|publisher=Cambridge University Press|year=2000|isbn=978-0-521-59180-5}}
*{{cite book|last1=Dõmõsi|first1=Pál|last2=Nehaniv|first2=Chrystopher L.|title=Algebraic Theory of Automata Networks|series=SIAM Monographs on Discrete Mathematics and Applications|publisher=SIAM|year=2004}}
*{{cite book|last1=Dummit|first1=David S.|last2=Foote|first2=Richard M.|title=Abstract Algebra|publisher=John Wiley & Sons|year=2004|edition=3rd|isbn=0-471-43334-9}}
*{{cite book|last=Fraleigh|first=John B.|title=A First Course in Abstract Algebra|publisher=Addison-Wesley|year=2003|edition=7th|isbn=978-0-321-15608-2}}
*{{cite book|last=Hall|first=Marshall|title=The Theory of Groups|publisher=Chelsea Publishing|location=Providence|year=1999|isbn=978-0-8218-1967-8}}
*{{cite book|last=Hungerford|first=Thomas|title=Algebra|series=Graduate Texts in Mathematics|publisher=Springer|year=2003}}
*{{cite book|last=Hungerford|first=Thomas|title=Abstract Algebra: An Introduction|publisher=Brooks/Cole Cengage Learning|year=2013}}
*{{cite book|last=Judson|first=Thomas W.|title=Abstract Algebra: Theory and Applications|year=2020|url=http://abstract.ups.edu/aata/aata.html}}
*{{cite book|last=Robinson|first=Derek J. S.|title=A Course in the Theory of Groups|volume=80|series=Graduate Texts in Mathematics|publisher=[[Springer-Verlag]]|year=1996|isbn=978-1-4612-6443-9|zbl=0836.20001|edition=2nd}}
*{{cite book|last=Thurston|first=William|author-link=William Thurston|title=Three-dimensional geometry and topology, Vol. 1|editor-last=Levy|editor-first=Silvio|series=Princeton Mathematical Series|publisher=Princeton University Press|year=1997|isbn=978-0-691-08304-9}}
*{{cite book|last=Bradley|first=C. J.|title=The mathematical theory of symmetry in solids : representation theory for point groups and space groups|publisher=Clarendon Press|location=Oxford New York|year=2010|isbn=978-0-19-958258-7|oclc=859155300}}
{{refend}}

== Further reading ==
* [[I. N. Herstein]], ''Topics in algebra.''  Second edition. Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. xi+388 pp.

== External links ==
* {{MathWorld|urlname=NormalSubgroup|title= normal subgroup}}
* [https://encyclopediaofmath.org/wiki/Normal_subgroup Normal subgroup in Springer's Encyclopedia of Mathematics]
* [http://www.math.uiuc.edu/~r-ash/Algebra/Chapter1.pdf Robert Ash: Group Fundamentals in ''Abstract Algebra. The Basic Graduate Year'']
* [http://gowers.wordpress.com/2011/11/20/normal-subgroups-and-quotient-groups Timothy Gowers, Normal subgroups and quotient groups]
* [http://math.ucr.edu/home/baez/normal.html John Baez, What's a Normal Subgroup?]

[[Category:Subgroup properties]]