Normal subgroup

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Definition

QUICK PHRASES: invariant under inner automorphisms, self-conjugate subgroup, same left and right cosets, kernel of a homomorphism, 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.

No. Shorthand A subgroup of a group is normal in it if... A subgroup of a group is normal in if ... Applications to... Additional comments
1 Homomorphism kernel it is the kernel of a homomorphism from the group. there is a homomorphism from to a group such that the kernel of is precisely . In other words, is the identity element of if and only if . proving normality In this case, we do not need to separately check that is a subgroup since the kernel of a homomorphism is automatically a subgroup.
2 Inner automorphism invariance it is invariant under all inner automorphisms. for all , . More explicitly, for all , we have . 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 conjugates in the whole group. for all in , . This definition also motivates the term self-conjugate subgroup for normal subgroup (which was used earlier).
4 Left/right cosets equal its left cosets are the same as its right cosets (that is, it commutes with every element of the group). for all in , . proving normality When we say , we only mean equality as sets. It is not necessary that for . That stronger condition defines central subgroup.
5 Union of conjugacy classes it is a union of conjugacy classes. is a union of conjugacy classes in
6 Commutator inside it contains its commutator with the whole group. the commutator (which coincides with the commutator ) is contained in . 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 for and for , we have and for all . 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.

This definition is presented using a tabular format. |View all pages with definitions in tabular format

Notation and terminology

For a subgroup of a group , we denote the normality of in by or Notations. In words, we say that is normal in or a normal subgroup of .

Equivalence of definitions

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 is conjugation by ) group acts as automorphisms by conjugation
(3) and (4) A direct manipulation of equations involving elements and subsets. For full proof, refer: 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. For full proof, refer: equivalence of conjugacy and commutator definitions of normality manipulating equations in groups

Copyable LaTeX

The following is LaTeX for a quick definition of normality.

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


This article is about a basic definition in group theory. The article text may, however, contain advanced material.
VIEW: Definitions built on this | Facts about this: (facts closely related to Normal subgroup, all facts related to Normal subgroup) |Survey articles about this | Survey articles about definitions built on this
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This article defines a subgroup property that is pivotal (viz important) among existing subgroup properties
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Importance

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

  • Normal subgroups are precisely the kernels of homomorphisms
  • Normal subgroups are precisely the subgroups invariant under inner automorphisms, 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 information: Ubiquity of normality

Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions |

Extreme examples

  1. The trivial subgroup is always normal. Further information: Trivial subgroup is normal
  2. Every group is normal as a subgroup of itself. Further information: Every group is normal in itself

Examples

  1. 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 groups, and the non-abelian ones are called Hamiltonian groups). Further information: abelian implies every subgroup is normal
  2. If is an internal direct product of subgroups and , both and are normal in . Further information: direct factor implies normal
  3. 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 information: subgroup-defining function value is characteristic, characteristic implies normal

Non-examples

Here are some examples of non-normal subgroups:

  1. In 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 information: S2 is not normal in S3
  2. More generally, in any dihedral group of degree at least , the two-element subgroup generated by a reflection is not normal. Further information: Two-element subgroup generated by reflection is not normal in dihedral group
  3. 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 the alternating group on five letters.

Subgroups satisfying the property

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

Here are some examples of subgroups in basic/important groups satisfying the property:

  Group part Subgroup part Quotient part
A3 in S3 Symmetric group:S3 Cyclic group:Z3 Cyclic group:Z2
Z2 in V4 Klein four-group Cyclic group:Z2 Cyclic group:Z2

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

  Group part Subgroup part Quotient part
A4 in S4 Symmetric group:S4 Alternating group:A4 Cyclic group:Z2
Center of dihedral group:D8 Dihedral group:D8 Cyclic group:Z2 Klein four-group
Center of quaternion group Quaternion group Cyclic group:Z2 Klein four-group
Center of special linear group:SL(2,3) Special linear group:SL(2,3) Cyclic group:Z2 Alternating group:A4
Center of special linear group:SL(2,5) Special linear group:SL(2,5) Cyclic group:Z2 Alternating group:A5
Cyclic maximal subgroup of dihedral group:D8 Dihedral group:D8 Cyclic group:Z4 Cyclic group:Z2
Cyclic maximal subgroups of quaternion group Quaternion group Cyclic group:Z4 Cyclic group:Z2
First agemo subgroup of direct product of Z4 and Z2 Direct product of Z4 and Z2 Cyclic group:Z2 Klein four-group
First omega subgroup of direct product of Z4 and Z2 Direct product of Z4 and Z2 Klein four-group Cyclic group:Z2
Klein four-subgroup of alternating group:A4 Alternating group:A4 Klein four-group Cyclic group:Z3
Klein four-subgroups of dihedral group:D8 Dihedral group:D8 Klein four-group Cyclic group:Z2
Non-characteristic order two subgroups of direct product of Z4 and Z2 Direct product of Z4 and Z2 Cyclic group:Z2 Cyclic group:Z4
Normal Klein four-subgroup of symmetric group:S4 Symmetric group:S4 Klein four-group Symmetric group:S3
Z4 in direct product of Z4 and Z2 Direct product of Z4 and Z2 Cyclic group:Z4 Cyclic group:Z2

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

  Group part Subgroup part Quotient part
Center of M16 M16 Cyclic group:Z4 Klein four-group
Center of central product of D8 and Z4 Central product of D8 and Z4 Cyclic group:Z4 Klein four-group
Center of dihedral group:D16 Dihedral group:D16 Cyclic group:Z2 Dihedral group:D8
Center of direct product of D8 and Z2 Direct product of D8 and Z2 Klein four-group Klein four-group
Center of nontrivial semidirect product of Z4 and Z4 Nontrivial semidirect product of Z4 and Z4 Klein four-group Klein four-group
Center of semidihedral group:SD16 Semidihedral group:SD16 Cyclic group:Z2 Dihedral group:D8
Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4 Nontrivial semidirect product of Z4 and Z4 Cyclic group:Z2 Quaternion group
Cyclic maximal subgroup of dihedral group:D16 Dihedral group:D16 Cyclic group:Z8 Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16 Semidihedral group:SD16 Cyclic group:Z8 Cyclic group:Z2
D8 in D16 Dihedral group:D16 Dihedral group:D8 Cyclic group:Z2
D8 in SD16 Semidihedral group:SD16 Dihedral group:D8 Cyclic group:Z2
Derived subgroup of M16 M16 Cyclic group:Z2 Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16 Dihedral group:D16 Cyclic group:Z4 Klein four-group
Derived subgroup of nontrivial semidirect product of Z4 and Z4 Nontrivial semidirect product of Z4 and Z4 Cyclic group:Z2 Direct product of Z4 and Z2
Direct product of Z4 and Z2 in M16 M16 Direct product of Z4 and Z2 Cyclic group:Z2
Klein four-subgroup of M16 M16 Klein four-group Cyclic group:Z4
Non-central Z4 in M16 M16 Cyclic group:Z4 Cyclic group:Z4
Q8 in SD16 Semidihedral group:SD16 Quaternion group Cyclic group:Z2
Q8 in central product of D8 and Z4 Central product of D8 and Z4 Quaternion group Cyclic group:Z2
SL(2,3) in GL(2,3) General linear group:GL(2,3) Special linear group:SL(2,3) Cyclic group:Z2
… further results

Subgroups dissatisfying the property

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

Here are some examples of subgroups in basic/important groups not satisfying the property:

  Group part Subgroup part Quotient part
S2 in S3 Symmetric group:S3 Cyclic group:Z2

Here are some some examples of subgroups in relatively less basic/important groups not satisfying the property:

  Group part Subgroup part Quotient part
A3 in A4 Alternating group:A4 Cyclic group:Z3
A3 in A5 Alternating group:A5 Cyclic group:Z3
A3 in S4 Symmetric group:S4 Cyclic group:Z3
A4 in A5 Alternating group:A5 Alternating group:A4
D8 in A6 Alternating group:A6 Dihedral group:D8
D8 in S4 Symmetric group:S4 Dihedral group:D8
Klein four-subgroup of alternating group:A5 Alternating group:A5 Klein four-group
Non-normal Klein four-subgroups of symmetric group:S4 Symmetric group:S4 Klein four-group
Non-normal subgroups of dihedral group:D8 Dihedral group:D8 Cyclic group:Z2
S2 in S4 Symmetric group:S4 Cyclic group:Z2
Subgroup generated by double transposition in symmetric group:S4 Symmetric group:S4 Cyclic group:Z2
Twisted S3 in A5 Alternating group:A5 Symmetric group:S3

Here are some examples of subgroups in even more complicated/less basic groups not satisfying the property:

  Group part Subgroup part Quotient part
2-Sylow subgroup of general linear group:GL(2,3) General linear group:GL(2,3) Semidihedral group:SD16
Non-normal subgroups of M16 M16 Cyclic group:Z2

Facts

Isomorphism theorems

Theorem name/number Statement
First isomorphism theorem If is a surjective homomorphism, and is the kernel of , then is a normal subgroup, and if is the quotient map, then there is a unique isomorphism such that .
Second isomorphism theorem (diamond isomorphism theorem) If are subgroups such that is contained in the normalizer of , then is normal in and .
Third isomorphism theorem If are groups with both normal in , then is normal in and .
Fourth isomorphism theorem (lattice isomorphism theorem) If is normal in , then there is a bijective correspondence between subgroups of and subgroups of containing , satisfying many nice conditions.

Metaproperties

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)
Metaproperty name Satisfied? Proof Statement with symbols
abelian-tautological subgroup property Yes abelian implies every subgroup is normal If and is abelian, then is normal in .
transitive subgroup property No Normality is not transitive We can have such that is normal in and is normal in but is not normal in .
trim subgroup property Yes Every group is normal in itself, trivial subgroup is normal trivial subgroup and whole group are both normal
strongly intersection-closed subgroup property Yes Normality is strongly intersection-closed If are all normal subgroups of a group , then the intersection of subgroups is also a normal subgroup of .
strongly join-closed subgroup property Yes Normality is strongly join-closed If are all normal subgroups of , then the join of subgroups is also a normal subgroup of .
quotient-transitive subgroup property Yes Normality is quotient-transitive If are groups such that is normal in and is normal in , then is normal in
intermediate subgroup condition Yes Normality satisfies intermediate subgroup condition If are groups such that is normal in , then is normal in .
transfer condition Yes Normality satisfies transfer condition If are groups such that is normal in , then we must have that normal in
image condition Yes Normality satisfies image condition If is a normal subgroup of and is a surjective homomorphism of groups, then normal in
inverse image condition Yes Normality satisfies inverse image condition If is a normal subgroup of and is a homomorphism of groups, then is normal in .
upper join-closed subgroup property Yes Normality is upper join-closed If , and are all subgroups of containing such that normal in each , then we must have that is normal in the join of subgroups .
commutator-closed subgroup property Yes Normality is commutator-closed If are both normal subgroups of a group , then the commutator normal in
centralizer-closed subgroup property Yes Normality is centralizer-closed If is a normal subgroup of , the centralizer is also normal in .
direct product-closed subgroup property Yes Normality is direct product-closed If is an indexing set, are groups, and are such that each is normal in the corresponding , then in the external direct product of the s, the subgroup given by the external direct product of the s is a normal subgroup.
This also implies that normality is a finite direct power-closed subgroup property.
strongly UL-intersection-closed subgroup property Yes Normality is strongly UL-intersection-closed If is an indexing set, is a group, and we have groups for each , such that is normal in , then is a normal subgroup of .
Arguesian subgroup property Yes Normality is Arguesian The collection of normal subgroups of a group form an Arguesian lattice.
lower central series condition Yes Normality satisfies lower central series condition Suppose is a normal subgroup of a group . Then, each lower central series member is a normal subgroup inside the corresponding lower central series member .
partition difference condition Yes Normality satisfies partition difference condition Suppose is a subgroup of a group , and has a partition as a union of . If all except possibly one of the s are normal subgroups of , then all the s are normal subgroups of .

Relation with other properties

This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied

Some of these can be found at:


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

Stronger properties

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

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions Comparison Collapse
characteristic subgroup invariant under all automorphisms characteristic implies normal normal not implies characteristic (see also list of examples) Center-fixing automorphism-invariant subgroup, Characteristic subgroup of direct factor, Characteristic-potentially characteristic subgroup, Cofactorial automorphism-invariant subgroup, IA-automorphism-invariant subgroup, Normal-extensible automorphism-invariant subgroup, Normal-potentially characteristic subgroup, Normal-potentially relatively characteristic subgroup, Semi-strongly image-potentially characteristic subgroup, Strongly image-potentially characteristic subgroup, Upper join of characteristic subgroups|FULL LIST, MORE INFO characteristic versus normal group in which every normal subgroup is characteristic
central factor inner automorphism of whole group is inner on subgroup central factor implies normal normal not implies central factor (see also list of examples) Conjugacy-closed normal subgroup, Locally inner automorphism-balanced subgroup, Normal subgroup whose center is contained in the center of the whole group, Normal subgroup whose focal subgroup equals its derived subgroup, SCAB-subgroup, Transitively normal subgroup|FULL LIST, MORE INFO central factor versus normal group in which every normal subgroup is a central factor
direct factor factor in internal direct product direct factor implies normal normal not implies direct factor (see also list of examples) Central factor, Characteristic subgroup of direct factor, Complemented central factor, Complemented normal subgroup, Complemented transitively normal subgroup, Conjugacy-closed normal subgroup, Direct factor over central subgroup, Endomorphism kernel, Intermediately endomorphism kernel, Join of finitely many direct factors, Join-transitively central factor, Locally inner automorphism-balanced subgroup, Normal AEP-subgroup, Normal subgroup having a 1-closed transversal, Normal subgroup in which every subgroup characteristic in the whole group is characteristic, Normal subgroup whose focal subgroup equals its derived subgroup, Powering-invariant normal subgroup, Quotient-powering-invariant subgroup, Right-quotient-transitively central factor, SCAB-subgroup … further results|FULL LIST, MORE INFO direct factor versus normal group in which every normal subgroup is a direct factor
central subgroup contained in the center central implies normal normal not implies central (see also list of examples) Abelian normal subgroup, Amalgam-characteristic subgroup, Amalgam-strictly characteristic subgroup, Center-fixing automorphism-invariant subgroup, Central factor, Class two normal subgroup, Commutator-in-center subgroup, Conjugacy-closed normal subgroup, Dedekind normal subgroup, Direct factor over central subgroup, Hereditarily normal subgroup, Join-transitively central factor, Nilpotent normal subgroup, SCAB-subgroup, Transitively normal subgroup|FULL LIST, MORE INFO normal subgroup versus central subgroup abelian group

Other less important properties that are stronger than normality:

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions Collapse
fully invariant subgroup invariant under all endomorphisms (via characteristic) (via characteristic)(see also list of examples) Characteristic subgroup, Finite direct power-closed characteristic subgroup, Fully invariant-potentially fully invariant subgroup, Image-potentially fully invariant subgroup, Injective endomorphism-invariant subgroup, Normal-potentially fully invariant subgroup, Normality-preserving endomorphism-invariant subgroup, Potentially fully invariant subgroup, Retraction-invariant characteristic subgroup, Retraction-invariant normal subgroup, Strictly characteristic subgroup|FULL LIST, MORE INFO group in which every normal subgroup is fully invariant
strictly characteristic subgroup invariant under surjective endomorphisms (via characteristic) (via characteristic) Characteristic subgroup|FULL LIST, MORE INFO
isomorph-free subgroup no other isomorphic subgroup (via characteristic) (via characteristic) (see also list of examples) Characteristic subgroup, Hall-relatively weakly closed subgroup, Injective endomorphism-invariant subgroup, Intermediately characteristic subgroup, Isomorph-automorphic normal subgroup, Isomorph-containing subgroup, Isomorph-normal characteristic subgroup, Isomorph-normal subgroup, Sub-isomorph-free subgroup|FULL LIST, MORE INFO group in which every normal subgroup is isomorph-free
isomorph-containing subgroup contains all isomorphic subgroups (via characteristic) (via characteristic) (see also list of examples) Characteristic subgroup, Injective endomorphism-invariant subgroup|FULL LIST, MORE INFO
injective endomorphism-invariant subgroup invariant under injective endomorphisms (via characteristic) (via characteristic) (see also list of examples) Characteristic subgroup|FULL LIST, MORE INFO
transitively normal subgroup any normal subgroup is normal in whole group (see also list of examples) | T-group
cocentral subgroup product with center is whole group cocentral implies normal normal not implies cocentral Central factor, Conjugacy-closed normal subgroup, Right-quotient-transitively central factor, SCAB-subgroup, Transitively normal subgroup|FULL LIST, MORE INFO abelian group
SCAB-subgroup subgroup-conjugating automorphism restricts to subgroup-conjugating automorphism of subgroup Transitively normal subgroup|FULL LIST, MORE INFO
conjugacy-closed normal subgroup conjugacy-closed subgroup and normal subgroup Normal subgroup whose center is contained in the center of the whole group, Normal subgroup whose focal subgroup equals its derived subgroup, Transitively normal subgroup|FULL LIST, MORE INFO
abelian-quotient subgroup contains the derived subgroup Nilpotent-quotient subgroup|FULL LIST, MORE INFO
complemented normal subgroup normal and permutably complemented (by definition) normal not implies permutably complemented Endomorphism kernel, Intermediately endomorphism kernel, Normal subgroup having a 1-closed transversal, Powering-invariant normal subgroup, Quotient-powering-invariant subgroup|FULL LIST, MORE INFO
endomorphism kernel occurs as the kernel of an endomorphism (by definition) normal not implies endomorphism kernel Powering-invariant normal subgroup, Quotient-powering-invariant subgroup|FULL LIST, MORE INFO
For a complete list of subgroup properties stronger than Normal subgroup, click here
STRONGER PROPERTIES SATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | trim | image condition | centralizer-closed |
STRONGER PROPERTIES DISSATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | inverse image condition | image condition | centralizer-closed |

Conjunction with other properties

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

Conjunction Other component of conjunction Intermediate notions Additional comments
conjugacy-closed normal subgroup conjugacy-closed subgroup Normal subgroup whose center is contained in the center of the whole group, Normal subgroup whose focal subgroup equals its derived subgroup, Transitively normal subgroup|FULL LIST, MORE INFO
transitively normal subgroup CEP-subgroup |
normal Sylow subgroup Sylow subgroup Complemented fully invariant subgroup, Complemented homomorph-containing subgroup, Complemented normal subgroup, Fully invariant subgroup, Homomorph-containing subgroup, Isomorph-free subgroup, Normal subgroup having no common composition factor with its quotient group, Normal subgroup having no nontrivial homomorphism from its quotient group, Normal subgroup having no nontrivial homomorphism to its quotient group, Normal-homomorph-containing subgroup, Order-normal subgroup, Order-unique subgroup, Sub-homomorph-containing subgroup|FULL LIST, MORE INFO
normal Hall subgroup Hall subgroup Complemented fully invariant subgroup, Complemented homomorph-containing subgroup, Complemented normal subgroup, Fully invariant subgroup, Homomorph-containing subgroup, Isomorph-free subgroup, Normal subgroup having no common composition factor with its quotient group, Normal subgroup having no nontrivial homomorphism from its quotient group, Normal subgroup having no nontrivial homomorphism to its quotient group, Normal-homomorph-containing subgroup, Order-normal subgroup, Order-unique subgroup, Sub-homomorph-containing subgroup|FULL LIST, MORE INFO

View a complete list of conjunctions of normality with subgroup properties
We are often also interested in the conjunction of normality with 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:

Conjunction Other component of conjunction Intermediate notions Additional comments
abelian normal subgroup abelian group Class two normal subgroup, Commutator-in-center subgroup, Dedekind normal subgroup, Nilpotent normal subgroup, Normal subgroup whose inner automorphism group is central in automorphism group, Solvable normal subgroup|FULL LIST, MORE INFO
cyclic normal subgroup cyclic group Abelian normal subgroup, Commutator-in-center subgroup, Dedekind normal subgroup, Hereditarily normal subgroup, Homocyclic normal subgroup, Nilpotent normal subgroup, Normal subgroup whose automorphism group is abelian, Normal subgroup whose inner automorphism group is central in automorphism group, Solvable normal subgroup, Transitively normal subgroup|FULL LIST, MORE INFO
solvable normal subgroup solvable group |
nilpotent normal subgroup nilpotent group Solvable normal subgroup|FULL LIST, MORE INFO
simple normal subgroup simple group |
perfect normal subgroup perfect group Normal subgroup whose focal subgroup equals its derived subgroup, Subgroup realizable as the commutator of the whole group and a subgroup, Upper join of characteristic subgroups|FULL LIST, MORE INFO
finite normal subgroup finite group Amalgam-characteristic subgroup, Amalgam-normal-subhomomorph-containing subgroup, Amalgam-strictly characteristic subgroup, Finitely generated normal subgroup, Join of finite normal subgroups, Normal closure of finite subset, Periodic normal subgroup, Potentially normal-subhomomorph-containing subgroup, Powering-invariant normal subgroup, Quotient-powering-invariant subgroup|FULL LIST, MORE INFO
periodic normal subgroup periodic group Amalgam-characteristic subgroup, Amalgam-normal-subhomomorph-containing subgroup, Amalgam-strictly characteristic subgroup, Potentially normal-subhomomorph-containing subgroup|FULL LIST, MORE INFO
finitely generated normal subgroup finitely generated group Normal closure of finite subset|FULL LIST, MORE INFO

View a complete list of conjunctions of normality with group properties
In some cases, we are interested in studying normal subgroups with the big group constrained to satisfy some group property. For instance:

Conjunction Property of the group Intermediate notions Additional comments
normal subgroup of finite group finite group Finite normal subgroup, Join of finite normal subgroups, Kernel of a characteristic action on an abelian group, Normal closure of finite subset, Normal subgroup of finitely generated group, Quotient-powering-invariant subgroup|FULL LIST, MORE INFO
normal subgroup of finitely generated group finitely generated group |
normal subgroup of periodic group periodic group Quotient-powering-invariant subgroup|FULL LIST, MORE INFO
normal subgroup of group of prime power order group of prime power order Normal subgroup of nilpotent group|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions Comparison Collapse
subnormal subgroup obtained using subordination operator on normality normal implies subnormal subnormal not implies normal (see also list of examples) 2-hypernormalized subgroup, 2-subnormal subgroup, 3-subnormal subgroup, 4-subnormal subgroup, Asymptotically fixed-depth join-transitively subnormal subgroup, Central factor of normal subgroup, Direct factor of normal subgroup, Finitarily hypernormalized subgroup, Finite-conjugate-join-closed subnormal subgroup, Intermediately join-transitively subnormal subgroup, Join of finitely many 2-subnormal subgroups, Join-transitively 2-subnormal subgroup, Join-transitively subnormal subgroup, Linear-bound join-transitively subnormal subgroup, Modular 2-subnormal subgroup, Modular subnormal subgroup, Normal subgroup of characteristic subgroup, Permutable 2-subnormal subgroup, Permutable subnormal subgroup, Subnormal-permutable subnormal subgroup|FULL LIST, MORE INFO contrasting subnormality of various depths T-group
permutable subgroup (also called quasinormal subgroup) permutes with every subgroup normal implies permutable permutable not implies normal Permutable 2-subnormal subgroup|FULL LIST, MORE INFO group in which every permutable subgroup is normal
2-subnormal subgroup normal subgroup of normal subgroup normality is not transitive (see also list of examples) 2-hypernormalized subgroup, Central factor of normal subgroup, Direct factor of normal subgroup, Join-transitively 2-subnormal subgroup, Modular 2-subnormal subgroup, Normal subgroup of characteristic subgroup, Permutable 2-subnormal subgroup, Transitively normal subgroup of normal subgroup|FULL LIST, MORE INFO contrasting subnormality of various depths T-group

Other less important properties that are weaker than normality:

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions Comparison Collapse
3-subnormal subgroup normal of normal of normal 2-subnormal subgroup, Normal subgroup of characteristic subgroup|FULL LIST, MORE INFO contrasting subnormality of various depths T-group
4-subnormal subgroup 2-subnormal of 2-subnormal 2-subnormal subgroup, Normal subgroup of characteristic subgroup|FULL LIST, MORE INFO contrasting subnormality of various depths T-group
ascendant subgroup Hypernormalized subgroup, Permutable subgroup, Subnormal subgroup|FULL LIST, MORE INFO
descendant subgroup Conjugate-permutable subgroup, Intersection of subnormal subgroups, Subnormal subgroup|FULL LIST, MORE INFO
serial subgroup Ascendant subgroup, Hypernormalized subgroup, Subnormal subgroup|FULL LIST, MORE INFO
hypernormalized subgroup iterated normalizers reach whole group 2-hypernormalized subgroup, Finitarily hypernormalized subgroup|FULL LIST, MORE INFO
finitarily hypernormalized subgroup iterated normalizers reach whole group in finite number of steps 2-hypernormalized subgroup|FULL LIST, MORE INFO
2-hypernormalized subgroup normalizer is normal |
pronormal subgroup conjugate to any conjugate in their join normal implies pronormal pronormal not implies normal (see also list of examples) Join-transitively pronormal subgroup, Subgroup whose join with any distinct conjugate is the whole group, Transfer-closed pronormal subgroup|FULL LIST, MORE INFO subnormal-to-normal and normal-to-characteristic group in which every pronormal subgroup is normal
weakly pronormal subgroup (via pronormal) Pronormal subgroup, Subgroup whose join with any distinct conjugate is the whole group|FULL LIST, MORE INFO (above)
paranormal subgroup (via pronormal) Pronormal subgroup|FULL LIST, MORE INFO (above)
weakly normal subgroup (via pronormal) NE-subgroup, Paranormal subgroup|FULL LIST, MORE INFO (above)
intermediately subnormal-to-normal subgroup (via pronormal) (see also list of examples) NE-subgroup, Paranormal subgroup, Polynormal subgroup, Pronormal subgroup, Weakly normal subgroup, Weakly pronormal subgroup|FULL LIST, MORE INFO (above)
conjugate-permutable subgroup permutes with its conjugate subgroups 2-hypernormalized subgroup, 2-subnormal subgroup, Permutable 2-subnormal subgroup, Permutable subgroup, Permutable subgroup of normal subgroup|FULL LIST, MORE INFO
modular subgroup Modular 2-subnormal subgroup, Modular subnormal subgroup, Permutable subgroup|FULL LIST, MORE INFO
automorph-permutable subgroup Normal subgroup of characteristic subgroup, Permutable subgroup|FULL LIST, MORE INFO
elliptic subgroup Permutable subgroup|FULL LIST, MORE INFO
For a complete list of subgroup properties weaker than Normal subgroup, click here
WEAKER PROPERTIES SATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | trim | inverse image condition | image condition |
WEAKER PROPERTIES DISSATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | intersection-closed |join-closed | inverse image condition | centralizer-closed |

Related operators

There are three important subgroup operators related to normality:

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 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:

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 the subset.

Analogues in other algebraic structures

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-automorphisms, 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

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

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

First-order description

This subgroup property is a first-order subgroup property, viz., it has a first-order description in the theory of groups.
View a complete list of first-order subgroup properties

The subgroup property of normality can be expressed in first-order language as follows: is normal in if and only if:

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

Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property
Function restriction expression is a normal subgroup of if ... This means that normality is ... Additional comments
inner automorphism function every inner automorphism of sends every element of to within the invariance property for inner automorphisms
inner automorphism endomorphism every inner automorphism of restricts to an endomorphism of the endo-invariance property for inner automorphisms; i.e., it is the invariance property for inner automorphism, which is a property stronger than the property of being an endomorphism
inner automorphism automorphism every inner automorphism of restricts to an automorphism of the auto-invariance property for inner automorphisms; i.e., it is the invariance property for inner automorphism, which is a group-closed property of automorphisms inner automorphism to automorphism is right tight for normality
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Relation implication expression

This subgroup property is a relation implication-expressible subgroup property: it can be defined and viewed using a relation implication expression
View other relation implication-expressible subgroup properties

Normality can be expressed in terms of the relation implication formalism as the relation implication operator with the left side being conjugate subgroups and the right side being equal subgroups:

Conjugate Equal

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

Variety formalism

This subgroup property can be described in the language of universal algebra, viewing groups as a variety of algebras
View other such subgroup properties

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-automorphisms. 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 ideals.

Testing

The testing problem

Further information: 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 is inside the subgroup. Thus, it reduces to the membership problem for the subgroup.

GAP command

This subgroup property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
The GAP command for testing this subgroup property is:IsNormal
The GAP command for listing all subgroups with this property is:NormalSubgroups
View subgroup properties testable with built-in GAP command|View subgroup properties for which all subgroups can be listed with built-in GAP commands | View subgroup properties codable in GAP
Learn more about using GAP

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

IsNormal (group, subgroup);

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:

NormalSubgroups(group);

History

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References

Textbook references

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

Book Page number Chapter and section Contextual information View
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347More info 82 formal definition,, and Theorem 6 giving equivalent formulations. Also, Page 80 (first use).
Topics in Algebra by I. N. HersteinMore info 50 Section 2.6 formal definition
Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189More info 41 Section 1.5 definition introduced through proposition Google Books
Algebra by Serge Lang, ISBN 038795385XMore info 14 definition in paragraph Google Books
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632More info 52 Point (4.8) formal definition, followed by equivalent definition-cum-proposition in (4.9)
Basic Algebra: Groups, Rings, and Fields by Paul Moritz Cohn, ISBN 1852335874, 13-digit ISBN 978-1852335878More info 27 Section 2.1 definition in paragraph. This book uses a right action convention for functions. Google Books
Schaum's outline of group theory by Benjamín Baumslag and Bruce Chandler, ISBN 0070041245, 13-digit ISBN 978-0070041240More info 111 formal definition after some motivating discussion Google Books

Graduate texts on group theory:

Book Page number Chapter and section Contextual information View
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261More info 6 definition in paragraph Google Books
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613More info 15 Proposition 1.3.15 definition introduced through proposition Google Books
An introduction to the theory of groups by Joseph J. Rotman, ISBN 0387942858, 13-digit ISBN 978-0387942858More info 30 formal definition Google Books

Online lecture notes