View source for Normal subgroup

{{perspectives}}
[[importance rank::1| ]]
==Definition==

{{quick phrase|[[quick phrase::invariant under inner automorphisms, self-conjugate subgroup]], [[quick phrase::same left and right cosets]], [[quick phrase::kernel of a homomorphism]], [[quick phrase::subgroup that is a union of conjugacy classes]]}}

===Equivalent definitions in tabular format===

Note that each of these definitions (except the first one, as noted) assumes that we ''already'' have a group and a subgroup. To prove normality using any of these definitions, we first need to check that we actually have a subgroup.

{| class="sortable" border="1"
! No. !! Shorthand !! A [[subgroup]] of a [[group]] is normal in it if... !! A subgroup <math>H</math> of a group <math>G</math> is normal in <math>G</math> if ... !! Applications to... !! Additional comments
|-
| 1 || homomorphism kernel || it is the [[Defining ingredient::kernel]] of a [[Defining ingredient::homomorphism]] from the group. || there is a [[homomorphism]] <math>\varphi</math> from <math>G</math> to a group <math>K</math> such that the [[kernel]] of <math>\varphi</math> is precisely <math>H\!</math>. In other words, <math>\varphi(x) \!</math> is the identity element of <math>K</math> if and only if <math>x \in H</math>. || [[Proving normality#Construct a homomorphism having it as kernel|proving normality]] || In this case, we do not need to separately check that <math>H</math> is a sub''group'' since the kernel of a homomorphism is automatically a subgroup.
|-
| 2 || inner automorphism invariance || it is invariant under all [[Defining ingredient::inner automorphism]]s. || for all <math>g \in G</math>, <math>gHg^{-1} \subseteq H</math>. More explicitly, for all <math>g \in G, h \in H</math>, we have <math>ghg^{-1} \in H</math>. || [[Proving normality#Verify invariance under inner automorphisms|proving normality]] || Thus, normality is the [[invariance property]] with respect to the property of an automorphism being inner. This definition also motivates the term ''invariant subgroup'' for normal subgroup (which was used earlier).
|- 
| 3 || equals conjugates || it equals each of its [[Defining ingredient::conjugate subgroups|conjugates]] in the whole group. || for all <math>g</math> in <math>G</math>, <math>gHg^{-1} = H</math>.  || || This definition also motivates the term ''self-conjugate subgroup'' for normal subgroup (which was used earlier).
|-
| 4 || left/right cosets equal || its [[Defining ingredient::left coset]]s are the same as its [[Defining ingredient::right coset]]s (that is, it commutes with every element of the group). || for all <math>g</math> in <math>G</math>, <math>gH = Hg</math>. || [[proving normality#Determine its left and right cosets|proving normality]] || When we say <math>gH = Hg</math>, we only mean equality as sets. It is ''not'' necessary that <math>gh = hg</math> for <math>h \in H</math>. That stronger condition defines [[central subgroup]].
|-
| 5 || union of conjugacy classes || it is a union of [[Defining ingredient::conjugacy class]]es. || <math>H</math> is a union of [[conjugacy class]]es in <math>G</math> || ||
|-
| 6 || commutator inside || it contains its [[commutator of two subgroups|commutator]] with the whole group. || the [[defining ingredient::commutator of two subgroups|commutator]] <math>[H,G]</math> (which coincides with the commutator <math>[G,H]</math>) is contained in <math>H</math>. || [[proving normality#Compute its commutator with the whole group|proving normality]] ||
|-
| 7 || conjugates of generating set inside || given a generating set for the whole group and a generating set for the subgroup, every conjugate of an element in the latter by an element in the former, as well as by its inverse, is in the subgroup. || given a generating set <math>A</math> for <math>G</math> and <math>B</math> for <math>H</math>, we have <math>aba^{-1} \in H</math> and <math>a^{-1}ba \in H</math> for all <math>a \in A, b\in B</math>. || [[normality testing problem]] || For finite groups, we need only check conjugates by elements in the generating set and not by their inverses.
|}

For more definitions, see [[nonstandard definitions of normal subgroup]].

{{tabular definition format}}

===Notation and terminology===

For a subgroup <math>H \!</math> of a group <math>G \!</math>, we denote the normality of <math>\! H</math> in <math>\! G</math> by <math>H \underline{\triangleleft} G</math> or <math>G \underline{\triangleright} H</math>{{subgroup-notation-page}}. In words, we say that <math>\! H</math> is ''normal in'' <math>\! G</math> or a ''normal subgroup of'' <math>\! G</math>.

===Equivalence of definitions===

{| class="sortable" border="1"
! Pair of definitions !! Explanation of equivalence !! More related information
|-
| (1) and (2) || [[Normal subgroup equals kernel of homomorphism]] || [[first isomorphism theorem]]
|-
| (2) and (3) || Follows from the more general fact that [[restriction of automorphism to subgroup invariant under it and its inverse is automorphism]], combined with the fact that the inverse of an inner automorphism is also an inner automorphism (in fact, the inverse of conjugation by <math>g</math> is conjugation by <math>g^{-1}</math>) || [[group acts as automorphisms by conjugation]]
|-
| (3) and (4) || A direct manipulation of equations involving elements and subsets. {{proofat|[[equivalence of conjugacy and coset definitions of normality]]}}.||[[manipulating equations in groups]]
|-
| (2) (or (3)) and (5) || A straightforward unraveling of the meaning of [[conjugacy class]] || 
|-
| (2) (or (3)) and (6) || A straightforward unraveling of the meaning of [[commutator]], along with a little bit of manipulation. {{proofat|[[equivalence of conjugacy and commutator definitions of normality]]}} || [[manipulating equations in groups]]
|}

===Copyable LaTeX===

The following is LaTeX for a quick definition of normality.

<pre>A subgroup $H$ of a group $G$ is termed a {\em normal subgroup} if $ghg^{-1} \in H$ for all $g \in G$ and $h \in H$.</pre>

{{basicdef}}
{{pivotal subgroup property}}

==Importance==

The notion of normal subgroup is important because of two main reasons:

* Normal subgroups are precisely the [[homomorphism of groups#kernel|kernel]]s of homomorphisms
* Normal subgroups are precisely the subgroups invariant under [[inner automorphism]]s, and for a [[group action]], the only relevant automorphisms of the acting group that correspond to symmetries of the set being acted upon, are inner automorphisms.

{{further|[[Ubiquity of normality]]}}

==Examples==

{{subgroup property see examples}}
===Extreme examples===

# The trivial subgroup is always normal. {{further|[[Trivial subgroup is normal]]}}
# Every group is normal as a subgroup of itself. {{further|[[Every group is normal in itself]]}}

===Examples===

# '''High occurrence example''': In an [[abelian group]], every subgroup is normal (there are non-abelian groups, such as the [[quaternion group]], where every subgroup is normal. Groups in which every subgroup is normal are called [[Dedekind group]]s, and the non-abelian ones are called Hamiltonian groups). {{further|[[abelian implies every subgroup is normal]]}}
# If <math>G</math> is an [[internal direct product]] of subgroups <math>H</math> and <math>K</math>, both <math>H</math> and <math>K</math> are normal in <math>G</math>. {{further|[[direct factor implies normal]]}}
# Every [[subgroup-defining function]] yields a normal subgroup (in fact, it yields a [[characteristic subgroup]]). For instance, the [[center]], [[derived subgroup]] and [[Frattini subgroup]] in any group are normal. {{further|[[subgroup-defining function value is characteristic]], [[characteristic implies normal]]}}

===Non-examples===

Here are some examples of non-normal subgroups:

# In [[symmetric group:S3|the symmetric group on three letters]], the subgroup [[S2 in S3]], i.e., the two-element subgroup generated by a transposition, is not normal (in fact, there are three such subgroups and they're all conjugate). {{further|[[S2 is not normal in S3]]}}
# More generally, in any [[dihedral group]] of degree at least <math>3</math>, the two-element subgroup generated by a ''reflection'' is not normal. {{further|[[Two-element subgroup generated by reflection is not normal in dihedral group]]}}
# '''Low occurrence example''': In a [[simple group]], no proper nontrivial subgroup is normal. Thus, any proper nontrivial subgroup of a simple group gives a counterexample. The smallest simple non-Abelian group is [[alternating group:A5|the alternating group on five letters]].

===Subgroups satisfying the property===

Here are examples of subgroups that ''satisfy'' the property of being normal:

{{subgroups satisfying property sorted by importance rank}}

===Subgroups dissatisfying the property===

Here are examples of subgroups that ''do not satisfy'' the property of being normal.

{{subgroups dissatisfying property sorted by importance rank}}

==Facts==

===Isomorphism theorems===

{| class="sortable" border="1"
! Theorem name/number !! Statement
|-
| [[First isomorphism theorem]] || If <math>\varphi:G \to H</math> is a surjective homomorphism, and <math>N</math> is the kernel of <math>\varphi</math>, '''then''' <math>N</math> is a normal subgroup, and if <math>\alpha:G \to G/N</math> is the quotient map, then there is a unique isomorphism <math>\psi:G/N \to H</math> such that <math>\psi \circ \alpha = \varphi</math>.
|-
| [[Second isomorphism theorem]] (diamond isomorphism theorem) || If <math>N,H \le G</math> are subgroups such that <math>H</math> is contained in the normalizer of <math>N</math>, '''then''' <math>N</math> is normal in <math>NH</math> and <math>NH/N \cong H/(H \cap N)</math>.
|-
| [[Third isomorphism theorem]] || If <math>H \le K \le G</math> are groups with both <math>H,K</math> normal in <math>G</math>, '''then''' <math>H</math> is normal in <math>K</math> and <math>(G/H)/(K/H) \cong G/K</math>.
|-
| [[Fourth isomorphism theorem]] (lattice isomorphism theorem) || If <math>H</math> is normal in <math>G</math>, '''then''' there is a bijective correspondence between subgroups of <math>G/H</math> and subgroups of <math>G</math> containing <math>H</math>, satisfying many nice conditions.
|}

==Metaproperties==

{{wikilocal-section}}

{| class="sortable" border="1"
!Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
|-
|[[satisfies metaproperty::abelian-tautological subgroup property]] || Yes || [[abelian implies every subgroup is normal]] || If <math>H \le G</math> and <math>G</math> is abelian, '''then''' <math>H</math> is normal in <math>G</math>.
|-
|[[dissatisfies metaproperty::transitive subgroup property]] || No || [[Normality is not transitive]] || We can have <math>H \le K \le G</math> such that <math>H</math> is normal in <math>K</math> and <math>K</math> is normal in <math>G</math> '''but''' <math>H</math> is not normal in <math>G</math>.
|-
| [[satisfies metaproperty::trim subgroup property]] || Yes || [[Every group is normal in itself]], [[trivial subgroup is normal]] || trivial subgroup and whole group are both normal
|-
|[[satisfies metaproperty::strongly intersection-closed subgroup property]] || Yes || [[Normality is strongly intersection-closed]] || If <math>H_i, i \in I</math> are all normal subgroups of a group <math>G</math>, '''then''' the [[intersection of subgroups]] <math>\bigcap_{i \in I} H_i</math> is also a normal subgroup of <math>G</math>.
|-
|[[satisfies metaproperty::strongly join-closed subgroup property]] || Yes || [[Normality is strongly join-closed]] || If <math>H_i, i \in I</math> are all normal subgroups of <math>G</math>, '''then''' the [[join of subgroups]] <math>\langle H_i \rangle_{i \in I}</math> is also a normal subgroup of <math>G</math>.
|-
|[[satisfies metaproperty::quotient-transitive subgroup property]] || Yes || [[Normality is quotient-transitive]] ||If <math>H \le K \le G</math> are groups such that <math>H</math> is normal in <math>G</math> and <math>K/H</math> is normal in <math>G/H</math>, '''then''' <math>K</math> is normal in <math>G</math>
|-
|[[satisfies metaproperty::intermediate subgroup condition]] || Yes || [[Normality satisfies intermediate subgroup condition]] || If <math>H \le K \le G</math> are groups such that <math>H</math> is normal in <math>G</math>, '''then''' <math>H</math> is normal in <math>K</math>.
|-
|[[satisfies metaproperty::transfer condition]] || Yes || [[Normality satisfies transfer condition]] || If <math>H, K \le G</math> are groups such that <math>H</math> is normal in <math>G</math>, '''then''' we must have that <math>H \cap K</math> normal in <math>K</math>
|-
|[[satisfies metaproperty::image condition]] || Yes || [[Normality satisfies image condition]] || If <math>H</math> is a normal subgroup of <math>G</math> and <math>\varphi:G \to K</math> is a surjective [[homomorphism of groups]], '''then''' <math>\varphi(H)</math> is normal in <math>K</math>.
|-
|[[satisfies metaproperty::inverse image condition]] || Yes || [[Normality satisfies inverse image condition]] || If <math>H</math> is a normal subgroup of <math>G</math> and <math>\varphi:K \to G</math> is a [[homomorphism of groups]], '''then''' <math>\varphi^{-1}(H)</math> is normal in <math>K</math>.
|-
|[[satisfies metaproperty::upper join-closed subgroup property]] || Yes || [[Normality is upper join-closed]] || If <math>H \le G</math>, and <math>K_i, i \in I</math> are all subgroups of <math>G</math> containing <math>H</math> such that <math>H</math> normal in each <math>K_i</math>, '''then''' we must have that <math>H</math> is normal in the [[join of subgroups]] <math>\langle K_i \rangle_{i \in I}</math>.
|-
|[[satisfies metaproperty::commutator-closed subgroup property]] || Yes || [[Normality is commutator-closed]] || If <math>H, K</math> are both normal subgroups of a group <math>G</math>, '''then''' the [[commutator of two subgroups|commutator]] <math>[H,K]</math> normal in <math>G</math>
|-
|[[satisfies metaproperty::centralizer-closed subgroup property]] || Yes || [[Normality is centralizer-closed]] || If <math>H</math> is a normal subgroup of <math>G</math>, the [[centralizer]] <math>C_G(H)</math> is also normal in <math>G</math>.
|-
|[[satisfies metaproperty::direct product-closed subgroup property]] || Yes || [[Normality is direct product-closed]] || If <math>I</math> is an indexing set, <math>G_i, i \in I</math> are groups, and <math>H_i, i \in I</math> are such that each <math>H_i</math> is normal in the corresponding <math>G_i</math>, '''then''' in the [[external direct product]] of the <math>G_i</math>s, the subgroup given by the external direct product of the <math>H_i</math>s is a normal subgroup.<br>This also implies that normality is a [[satisfies metaproperty::finite direct power-closed subgroup property]].
|-
|[[satisfies metaproperty::strongly UL-intersection-closed subgroup property]] || Yes || [[Normality is strongly UL-intersection-closed]] || If <math>I</math> is an indexing set, <math>G</math> is a group, and we have groups <math>H_i \le K_i \le G</math> for each <math>i \in I</math>, such that <math>H_i</math> is normal in <math>K_i</math>, '''then''' <math>\bigcap_{i \in I} H_i</math> is a normal subgroup of <math>\bigcap_{i \in I} K_i</math>.
|-
|[[satisfies metaproperty::Arguesian subgroup property]] || Yes || [[Normality is Arguesian]] || The collection of normal subgroups of a group form an Arguesian lattice.
|-
| [[satisfies metaproperty::lower central series condition]] || Yes || [[Normality satisfies lower central series condition]] || Suppose <math>H</math> is a normal subgroup of a group <math>G</math>. Then, each lower central series member <math>\gamma_k(H)</math> is a normal subgroup inside the corresponding lower central series member <math>\gamma_k(G)</math>.
|-
| [[satisfies metaproperty::partition difference condition]] || Yes || [[Normality satisfies partition difference condition]] || Suppose <math>H</math> is a subgroup of a group <math>G</math>, and <math>H</math> has a nontrivial [[partition of a group|partition]] as a union of <math>H_i, i \in I</math> (nontrivial means that <math>I</math> has size more than one). If all except possibly one of the <math>H_i</math>s are normal subgroups of <math>G</math>, then all the <math>H_i</math>s are normal subgroups of <math>G</math>.
|-
| [[dissatisfies metaproperty::conditionally lattice-determined subgroup property]] || No || [[No subgroup property between normal Sylow and subnormal or between Sylow retract and retract is conditionally lattice-determined]] || It is possible to have a group <math>G</math>, an automorphism <math>\varphi</math> of the [[lattice of subgroups]], and a normal subgroup <math>H</math> of <math>G</math> such that <math>\varphi(H)</math> is not normal.
|}

==Relation with other properties==

{{pivotalproperty}}

Some of these can be found at:

* {{#ask: [[Variation of::Normal subgroup]]|limit = 0|searchlabel = Variations of normal subgroup}}
* {{#ask: [[Opposite of::Normal subgroup]]|limit = 0|searchlabel = Opposites of normal subgroup}}
* {{#ask: [[Analogue of::Normal subgroup]]|limit = 0|searchlabel = Analogues of normal subgroup}}


To get a broad overview, check out the survey articles:

* [[Varying normality]]
* [[Subnormal-to-normal and normal-to-characteristic]]
* [[Between normal and characteristic and beyond]]
* [[Between normal and subnormal and beyond]]
* [[Contrasting subnormality of various depths]]

===Stronger properties===

The most important stronger property is [[characteristic subgroup]]. See the table below for many stronger properties and the way they're related:

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions !! Comparison !! Collapse
|-
| [[weaker than::characteristic subgroup]] || invariant under all [[automorphism]]s || [[characteristic implies normal]] || [[normal not implies characteristic]] {{strictness examples for subgroup property|normal subgroup|characteristic subgroup}} || {{intermediate notions short|normal subgroup|characteristic subgroup}} || [[characteristic versus normal]] || [[group in which every normal subgroup is characteristic]]
|-
| [[weaker than::central factor]] || [[inner automorphism]] of whole group is inner on subgroup || [[central factor implies normal]] || [[normal not implies central factor]] {{strictness examples|normal subgroup|central factor}} || {{intermediate notions short|normal subgroup|central factor}} || [[central factor versus normal]] || [[group in which every normal subgroup is a central factor]]
|-
| [[weaker than::direct factor]] || factor in [[internal direct product]] || [[direct factor implies normal]] || [[normal not implies direct factor]] {{strictness examples for subgroup property|normal subgroup|direct factor}} || {{intermediate notions short|normal subgroup|direct factor}} || [[direct factor versus normal]] || [[group in which every normal subgroup is a direct factor]]
|-
| [[weaker than::central subgroup]] || contained in the [[center]] || [[central implies normal]] || [[normal not implies central]] {{strictness examples for subgroup property|normal subgroup|central subgroup}} || {{intermediate notions short|normal subgroup|central subgroup}} || [[normal subgroup versus central subgroup]] || [[abelian group]]
|}

Other less important properties that are stronger than normality:

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions !! Collapse
|-
| [[weaker than::fully invariant subgroup]] || invariant under all [[endomorphism]]s || (via characteristic) || (via characteristic){{strictness examples|normal subgroup|fully invariant subgroup}} || {{intermediate notions short|normal subgroup|fully invariant subgroup}} || [[group in which every normal subgroup is fully invariant]]
|-
| [[weaker than::strictly characteristic subgroup]] || invariant under [[surjective endomorphism]]s || (via characteristic) || (via characteristic) {{strictness examples|normal subgroup|strictly characteristic subgroup}} || {{intermediate notions short|normal subgroup|strictly characteristic subgroup}} || 
|-
| [[weaker than::isomorph-free subgroup]] || no other isomorphic subgroup || (via characteristic) || (via characteristic) {{strictness examples|normal subgroup|isomorph-free subgroup}} || {{intermediate notions short|normal subgroup|isomorph-free subgroup}} || [[group in which every normal subgroup is isomorph-free]]
|-
| [[weaker than::isomorph-containing subgroup]] || contains all isomorphic subgroups || (via characteristic) || (via characteristic) {{strictness examples|normal subgroup|isomorph-containing subgroup}} || {{intermediate notions short|normal subgroup|isomorph-containing subgroup}} || 
|-
| [[weaker than::injective endomorphism-invariant subgroup]] || invariant under injective endomorphisms || (via characteristic) || (via characteristic) {{strictness examples|normal subgroup|injective endomorphism-invariant subgroup}} || {{intermediate notions short|normal subgroup|injective endomorphism-invariant subgroup}} || 
|-
| [[weaker than::transitively normal subgroup]] || any normal subgroup is normal in whole group || || {{strictness examples for subgroup property|normal subgroup|transitively normal subgroup}} || {{intermediate notions short|normal subgroup|transitively normal subgroup}} || [[T-group]]
|-
| [[weaker than::cocentral subgroup]] || product with center is whole group || [[cocentral implies normal]] || [[normal not implies cocentral]] || {{intermediate notions short|normal subgroup|cocentral subgroup}} || [[abelian group]]
|-
| [[weaker than::SCAB-subgroup]] || [[subgroup-conjugating automorphism]] restricts to subgroup-conjugating automorphism of subgroup || || || {{intermediate notions short|normal subgroup|SCAB-subgroup}} || 
|-
| [[weaker than::conjugacy-closed normal subgroup]] || [[conjugacy-closed subgroup]] and normal subgroup || || || {{intermediate notions short|normal subgroup|conjugacy-closed normal subgroup}} || 
|-
| [[weaker than::abelian-quotient subgroup]] || contains the [[derived subgroup]] || || || {{intermediate notions short|normal subgroup|abelian-quotient subgroup}} ||
|-
| [[weaker than::complemented normal subgroup]] || normal and [[permutably complemented subgroup|permutably complemented]] || (by definition) || [[normal not implies permutably complemented]] || {{intermediate notions short|normal subgroup|complemented normal subgroup}} ||
|-
| [[weaker than::endomorphism kernel]] || occurs as the [[kernel]] of an [[endomorphism]] || (by definition) || [[normal not implies endomorphism kernel]] || {{intermediate notions short|normal subgroup|endomorphism kernel}} || 
|}

{{stronger subgroup properties filtered by metaproperty}}

===Conjunction with other properties===

Important conjunctions of normality with other subgroup properties are in the table below:

{| class="sortable" border="1"
! Conjunction !! Other component of conjunction !! Intermediate notions !! Additional comments
|-
| [[Weaker than::conjugacy-closed normal subgroup]] || [[conjugacy-closed subgroup]] || {{intermediate notions short|normal subgroup|conjugacy-closed normal subgroup}} ||
|-
| [[Weaker than::transitively normal subgroup]] || [[CEP-subgroup]] || {{intermediate notions short|normal subgroup|transitively normal subgroup}} ||
|-
| [[Weaker than::normal Sylow subgroup]] || [[Sylow subgroup]] || {{intermediate notions short|normal subgroup|normal Sylow subgroup}} ||
|-
| [[Weaker than::normal Hall subgroup]] || [[Hall subgroup]] || {{intermediate notions short|normal subgroup|normal Hall subgroup}} ||
|}

''{{#ask:[[Category:Subgroup property conjunctions]][[Conjunction involving::Normal subgroup]]|limit = 0| searchlabel = View a complete list of conjunctions of normality with subgroup properties}}''
<br>
We are often also interested in the conjunction of normality with [[group property|group properties]]. By this, we mean the subgroup property of being normal as a subgroup and having the given group property as an abstract group. Examples are in the table below:
{| class="sortable" border="1"
! Conjunction !! Other component of conjunction !! Intermediate notions !! Additional comments
|-
| [[Weaker than::abelian normal subgroup]] || [[abelian group]]  || {{intermediate notions short|normal subgroup|abelian normal subgroup}} ||
|-
| [[Weaker than::cyclic normal subgroup]] || [[cyclic group]] || {{intermediate notions short|normal subgroup|cyclic normal subgroup}} ||
|-
| [[Weaker than::solvable normal subgroup]] || [[solvable group]] || {{intermediate notions short|normal subgroup|solvable normal subgroup}} ||
|-
| [[Weaker than::nilpotent normal subgroup]] || [[nilpotent group]] || {{intermediate notions short|normal subgroup|nilpotent normal subgroup}} ||
|-
| [[Weaker than::simple normal subgroup]] || [[simple group]] || {{intermediate notions short|normal subgroup|simple normal subgroup}} ||
|-
|[[Weaker than::perfect normal subgroup]] || [[perfect group]] || {{intermediate notions short|normal subgroup|perfect normal subgroup}} ||
|-
| [[Weaker than::finite normal subgroup]] || [[finite group]] || {{intermediate notions short|normal subgroup|finite normal subgroup}} ||
|-
|[[Weaker than::periodic normal subgroup]] || [[periodic group]] || {{intermediate notions short|normal subgroup|periodic normal subgroup}} ||
|-
|[[Weaker than::finitely generated normal subgroup]] || [[finitely generated group]] || {{intermediate notions short|normal subgroup|finitely generated normal subgroup}} ||
|}

''{{#ask:[[Category:Conjunctions of group property with subgroup property]][[Conjunction involving::Normal subgroup]]|limit = 0| searchlabel = View a complete list of conjunctions of normality with group properties}}''
<br>
In some cases, we are interested in studying normal subgroups with the big group constrained to satisfy some [[group property]]. For instance:

{| class="sortable" border="1"
! Conjunction !! Property of the group !! Intermediate notions !! Additional comments
|-
| [[Weaker than::normal subgroup of finite group]] || [[finite group]] || {{intermediate notions short|normal subgroup|normal subgroup of finite group}} || 
|-
| [[Weaker than::normal subgroup of finitely generated group]] || [[finitely generated group]] || {{intermediate notions short|normal subgroup|normal subgroup of finitely generated group}} ||
|-
| [[weaker than::normal subgroup of periodic group]] || [[periodic group]] || {{intermediate notions short|normal subgroup|normal subgroup of periodic group}} ||
|-
| [[weaker than::normal subgroup of group of prime power order]] || [[group of prime power order]] || {{intermediate notions short|normal subgroup|normal subgroup of group of prime power order}} ||
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions !! Comparison !! Collapse
|-
| [[Stronger than::subnormal subgroup]] ||obtained using [[subordination operator]] on normality || [[normal implies subnormal]] || [[subnormal not implies normal]] {{strictness examples for subgroup property|subnormal subgroup|normal subgroup}}|| {{intermediate notions short|subnormal subgroup|normal subgroup}} || [[contrasting subnormality of various depths]] || [[T-group]]
|-
| [[Stronger than::permutable subgroup]] (also called '''quasinormal subgroup''') || [[permuting subgroups|permutes]] with every subgroup || [[normal implies permutable]] || [[permutable not implies normal]] {{strictness examples for subgroup property|permutable subgroup|normal subgroup}} || {{intermediate notions short|permutable subgroup|normal subgroup}} || || [[group in which every permutable subgroup is normal]]
|-
| [[Stronger than::2-subnormal subgroup]] || normal subgroup of normal subgroup || || [[normality is not transitive]] {{strictness examples for subgroup property|2-subnormal subgroup|normal subgroup}} || {{intermediate notions short|2-subnormal subgroup|normal subgroup}} || [[contrasting subnormality of various depths]] || [[T-group]]
|}

Other less important properties that are weaker than normality:

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions !! Comparison !! Collapse
|-
| [[Stronger than::3-subnormal subgroup]] || normal of normal of normal || || || {{intermediate notions short|3-subnormal subgroup|normal subgroup}} || [[contrasting subnormality of various depths]] || [[T-group]]
|-
| [[Stronger than::4-subnormal subgroup]] || 2-subnormal of 2-subnormal || || || {{intermediate notions short|4-subnormal subgroup|normal subgroup}} || [[contrasting subnormality of various depths]] || [[T-group]]
|-
| [[Stronger than::ascendant subgroup]] || || || || {{intermediate notions short|ascendant subgroup|normal subgroup}} || || 
|-
| [[Stronger than::descendant subgroup]] || || || || {{intermediate notions short|descendant subgroup|normal subgroup}} || || 
|-
| [[Stronger than::serial subgroup]] || || || || {{intermediate notions short|serial subgroup|normal subgroup}} || ||
|-
| [[Stronger than::hypernormalized subgroup]] || iterated [[normalizer]]s reach whole group || || {{strictness examples for subgroup property|hypernormalized subgroup|normal subgroup}} || {{intermediate notions short|hypernormalized subgroup|normal subgroup}} || ||
|-
| [[Stronger than::finitarily hypernormalized subgroup]] || iterated normalizers reach whole group in finite number of steps || || || {{intermediate notions short|finitarily hypernormalized subgroup|normal subgroup}} || ||
|-
| [[Stronger than::2-hypernormalized subgroup]] || [[normalizer]] is normal || || || {{intermediate notions short|2-hypernormalized subgroup|normal subgroup}} || ||
|-
| [[Stronger than::pronormal subgroup]] || conjugate to any conjugate in their join || [[normal implies pronormal]] || [[pronormal not implies normal]] {{strictness examples for subgroup property|pronormal subgroup|normal subgroup}} || {{intermediate notions short|pronormal subgroup|normal subgroup}} || [[subnormal-to-normal and normal-to-characteristic]] || [[group in which every pronormal subgroup is normal]]
|-
| [[Stronger than::weakly pronormal subgroup]] || || (via pronormal) ||  || {{intermediate notions short|weakly pronormal subgroup|normal subgroup}} || (above) ||
|-
| [[Stronger than::paranormal subgroup]] ||  || (via pronormal) || || {{intermediate notions short|paranormal subgroup|normal subgroup}} || (above) ||
|-
| [[Stronger than::weakly normal subgroup]] ||  || (via pronormal) ||  || {{intermediate notions short|weakly normal subgroup|normal subgroup}} || (above) ||
|-
|[[Stronger than::intermediately subnormal-to-normal subgroup]] ||  || (via pronormal) || {{strictness examples for subgroup property|intermediately subnormal-to-normal subgroup|normal subgroup}} || {{intermediate notions short|intermediately subnormal-to-normal subgroup|normal subgroup}} || (above) ||
|-
|[[Stronger than::conjugate-permutable subgroup]] || [[permuting subgroups|permutes]] with its [[conjugate subgroups]] || || || {{intermediate notions short|conjugate-permutable subgroup|normal subgroup}} || ||
|-
|[[Stronger than::modular subgroup]] || || || || {{intermediate notions short|modular subgroup|normal subgroup}} || ||
|-
|[[Stronger than::automorph-permutable subgroup]] || || || || {{intermediate notions short|automorph-permutable subgroup|normal subgroup}} || ||
|-
|[[Stronger than::elliptic subgroup]] || || || || {{intermediate notions short|elliptic subgroup|normal subgroup}} || ||
|}

{{weaker subgroup properties filtered by metaproperty}}

===Related operators===

There are three important subgroup operators related to normality:

{| class="sortable" border="1"
! Operator !! What it does !! How normal subgroups are related to it
|-
| [[Normal core]] || This takes a subgroup and outputs the largest normal subgroup inside it, which is also the intersection of all its [[conjugate subgroups]] || The operator is idempotent (doing it twice is the same as doing it once) and normal subgroups are precisely the subgroups that are invariant under it, and hence also precisely the subgroups that can arise from it.
|-
| [[Normal closure]] || This takes a subgroup and outputs the smallest normal subgroup containing it, which is also the [[join of subgroups|join]] of all its [[conjugate subgroups]] || The operator is idempotent (doing it twice is the same as doing it once) and normal subgroups are precisely the subgroups that are invariant under it, and hence also precisely the subgroups that can arise from it.
|-
| [[Normalizer]] || This takes a subgroup and outputs the largest subgroup within which it is normal || A subgroup is normal if and only if its normalizer is the whole group
|}

Other operators involve composing these in different ways, for instance:

* [[Normal core of normalizer]]
* [[Hypernormalizer]]

Closely related to [[normal closure]] is the [[normal subgroup generated by a subset]], which is defined as the smallest normal subgroup containing the subset, and is the normal closure of the [[subgroup generated by a subset|subgroup generated]] by the subset.

===Analogues in other algebraic structures===

{| class="sortable" border="1"
! Algebraic structure !! Analogue of normal subgroup in that structure !! Definition !! Nature of analogy with normal subgroup
|-
| [[Lie ring]] || [[Ideal of a Lie ring]] || A subring of a Lie ring whose Lie bracket with any element of the Lie ring is in the subring. || Precisely the kernels of homomorphisms, play analogous roles in isomorphism theorems. Also, precisely the subrings invariant under inner derivations. Also, the Lazard correspondence maps ideals of the Lazard Lie ring to normal subgroups of the group.
|-
| ideal-determined variety of algebras (universal algebra)|| ideal in that variety || An ideal-determined variety of algebras with zero is a variety where every ideal occurs as the inverse image of zero under some homomorphism and this completely determines the fibers of the homomorphism. Also, the various isomorphism theorems hold with suitable modifications. || [[variety of groups is ideal-determined]], with the ideals being the normal subgroups.
|-
| [[variety of algebras]] (universal algebra) ||  [[I-automorphism-invariant subalgebra]] || invariant under [[I-automorphism]]s, which are the automorphism described by formulas that universally give automorphisms. || [[inner automorphisms are I-automorphisms in the variety of groups]], so the corresponding invariant subalgebras are subgroups.
|-
| [[loop]] || [[normal subloop]] || commutes with every element, associates with every pair of elements || when the loop is a group, then normal subloop = normal subgroup. Also, normal subloops are kernels of homomorphisms and the isomorphism theorems hold.
|-
| [[hypergroup]] || [[normal subhypergroup]] || ||
|-
| [[semigroup]] || [[left-normal subsemigroup]], [[right-normal subsemigroup]] || ||
|}

==Effect of property operators==

{| class="sortable" border="1"
! Operator !! Meaning !! Result of application !! Proof
|-
| [[left transiter]] || if big group is normal in a bigger group, so is subgroup || [[characteristic subgroup]] || [[left transiter of normal is characteristic]]
|-
| [[right transiter]] || every normal subgroup of subgroup is normal in whole group || [[transitively normal subgroup]] || by definition
|-
| [[subordination operator]] || normal subgroup of normal subgroup of ... of normal subgroup || [[subnormal subgroup]] || by definition 
|-
| [[hereditarily operator]] || every subgroup of it is normal || [[hereditarily normal subgroup]] || by definition 
|-
| [[upward-closure operator]] || every subgroup containing it is normal || [[upward-closed normal subgroup]] || by definition
|-
| [[maximal proper operator]] || proper normal subgroup contained in no other proper normal subgroup || [[maximal normal subgroup]]|| by definition
|-
| [[minimal operator]] ||  nontrivial normal subgroup containing no other nontrivial normal subgroup || [[minimal normal subgroup]]|| by definition
|}

==Formalisms==

{{wikilocal-section}}

{{first-order subgroup property}}

The subgroup property of normality can be expressed in first-order language as follows: <math>H</math> is normal in <math>G</math> if and only if:

<math>\forall g \in G, h \in H: \ ghg^{-1} \in H</math>

This is in fact a universally quantified expression of Fraisse rank 1.

{{frexp}}

{| class="sortable" border="1"
! Function restriction expression !! <math>H</math> is a normal subgroup of <math>G</math> if ... !! This means that normality is ... !! Additional comments
|-
| {{invariance-short|inner automorphism}} || 
|-
| {{endo-invariance-short|inner automorphism}} ||
|-
| {{auto-invariance-short|inner automorphism}} || [[inner automorphism to automorphism is right tight for normality]]
|}

<toggledisplay>
{| class="sortable" border="1"
! Function restriction expression !! <math>H</math> is a normal subgroup of <math>G</math> if ... !! This means that normality is ... !! Additional comments
|-
| {{invariance-short|class-preserving automorphism}} ||
|-
| {{endo-invariance-short|class-preserving automorphism}} ||
|-
| {{auto-invariance-short|class-preserving automorphism}} || class-preserving automorphism to automorphism is right tight for normality
|-
| {{invariance-short|subgroup-conjugating automorphism}} ||
|-
| {{endo-invariance-short|subgroup-conjugating automorphism}} ||
|-
| {{auto-invariance-short|subgroup-conjugating automorphism}} ||
|-
| {{invariance-short|normal automorphism}} ||
|-
| {{endo-invariance-short|normal automorphism}} ||
|-
| {{auto-invariance-short|normal automorphism}} || [[normal automorphism to automorphism is left tight for normality]] (also right tight)
|-
| {{invariance-short|weakly normal automorphism}} || 
|-
| {{endo-invariance-short|weakly normal automorphism}} ||
|-
| {{invariance-short|monomial automorphism}} ||
|-
| {{endo-invariance-short|monomial automorphism}} ||
|-
| {{invariance-short|strong monomial automorphism}} ||
|-
| {{endo-invariance-short|strong monomial automorphism}} ||
|-
| {{auto-invariance-short|strong monomial automorphism}} ||
|}

{{quotation|For more function restriction expressions for normality, check out [[Nonstandard definitions of normal subgroup#Definition via function restriction expression]].}}
</toggledisplay>
{{riexp}}

Normality can be expressed in terms of the [[relation implication formalism]][[Category: Properties expressible using the relation implication formalism]] as the relation implication operator with the left side being [[conjugate subgroups]] and the right side being '''equal subgroups''':

Conjugate <math>\implies</math> Equal

In other words, a subgroup is normal if any subgroup related to it by being conjugate is in fact equal to it.

{{variety-expressible subgroup property}}

There are two somewhat different ways of expressing the notion of normality in the language of varieties:

* In the [[variety of groups]], the normal subgroups are precisely the subalgebras invariant under all the [[I-automorphism]]s. An I-automorphism is an automorphism that can be expressed using a ''formula'' guaranteed to give an automorphism. This definition of normal subgroup follows from the fact that for groups, inner automorphisms are precisely the I-automorphisms.
* Treating the variety of groups as a [[variety of algebras with zero]], the normal subgroups are precisely the [[ideal in a variety with zero|ideal]]s.

==Testing==

===The testing problem===
{{further|[[Normality testing problem]]}}

Given generating sets for a group and a subgroup, the problem of determining whether the subgroup is normal in the group reduces to the problem of testing whether the conjugate of any generator of the subgroup by any generator of the group, and by the inverse of the generator, is inside the subgroup. Thus, it reduces to the [[membership problem]] for the subgroup.

{{GAP command for subgroup property|
test = IsNormal|
list = NormalSubgroups}}

The GAP syntax for testing whether a subgroup is normal in a group is:

<tt>IsNormal (group, subgroup);</tt>

where subgroup and group may be defined on the spot in terms of generators (described as permutations) or may refer to things previously defined.

GAP can also be used to list all normal subgroups of a given group, using the command:

<tt>NormalSubgroups(group);</tt>

==History==
<toggledisplay>
===Origin of the concept===

The notion of normal subgroup dates to an era before group theory began formally. Normal subgroups arose as subgroups for which the quotient group is well-defined.

Normal subgroups were earlier termed ''invariant subgroups'' (because they were [[invariance property|invariant]] under [[inner automorphism]]s) and also termed ''self-conjugate subgroups'' (because a normal subgroup is precisely a subgroup that equals every conjugate).

===Origin of the term===

{{term introduced by|Galois}}

The term ''normal subgroup'' arose because, under the Galois correspondence established by the fundamental theorem of Galois theory between subgroups and subfields, the normal subgroups corresponded precisely to the subfields that were normal extensions over the base field.
</toggledisplay>

==References==

===Textbook references===

Advanced undergraduate/beginning graduate algebra texts that include group theory:

{| class="sortable" border="1"
! Book !! Page number !! Chapter and section !! Contextual information !! View
|-
| {{booklink-defined-tabular|DummitFoote|82||formal definition}} and Theorem 6 giving equivalent formulations. Also, Page 80 (first use). || 
|-
| {{booklink-defined-tabular|Herstein|50|Section 2.6|formal definition}} || 
|-
| {{booklink-defined-tabular|Hungerford|41|Section 1.5|definition introduced through proposition}} || [http://books.google.com/books?id=t6N_tOQhafoC&pg=PA41 Google Books]
|-
| {{booklink-defined-tabular|Lang|14||definition in paragraph}} || [http://books.google.com/books?id=Fge-BwqhqIYC&pg=PA14 Google Books]
|-
| {{booklink-defined-tabular|Artin|52|Point (4.8)|formal definition}}, followed by equivalent definition-cum-proposition in (4.9) ||
|-
| {{booklink-defined-tabular|CohnBasicAlgebra|27|Section 2.1|definition in paragraph}}. This book uses a right action convention for functions. || [http://books.google.com/books?id=VESm0MJOiDQC&pg=PA27 Google Books]
|-
| {{booklink-defined-tabular|SchaumsGT|111||formal definition}} after some motivating discussion || [http://books.google.com/books?id=gZ-yQ1wMaGsC&pg=PA111 Google Books]
|}

Graduate texts on group theory:

{| class="sortable" border="1"
! Book !! Page number !! Chapter and section !! Contextual information !! View
|-
| {{booklink-defined-tabular|AlperinBell|6||definition in paragraph}} || [http://books.google.com/books?id=fYWkPNiK0wwC&pg=PA6 Google Books]
|-
| {{booklink-defined-tabular|RobinsonGT|15|Proposition 1.3.15|definition introduced through proposition}} || [http://books.google.com/books?id=Zl3mGxwUtmEC&pg=PA15 Google Books]
|-
| {{booklink-defined-tabular|RotmanGT|30||formal definition}} || [http://books.google.com/books?id=lYrsiaHSHKcC&pg=PA30 Google Books]
|}

===Online lecture notes===

* [http://www.jmilne.org/math/CourseNotes/gt.html J.S. Milne's course notes], Section 1.7, Page 15 (both the A4 and the letter versions)
* [http://www.extension.harvard.edu/openlearning/math222/files/notes/L4-N.pdf Lecture notes for Benedict Gross's lecture 4], Page 2-3 and more. You can also access the video lectures and other material for the course through the [http://www.extension.harvard.edu/openlearning/math222/ course main page].
* [http://www.solitaryroad.com/c301.html Lecture notes on basic group theory] by James Miller, part of the solitaryroad.com website