Have questions about this topic? Check out Questions:Normal subgroup  it may already contain your question.
Definition
QUICK PHRASES: invariant under inner automorphisms, selfconjugate subgroup, same left and right cosets, kernel of a homomorphism, subgroup that is a union of conjugacy classes
Equivalent definitions in tabular format
Six equivalent definitions of normality are listed below. Note that each of these definitions (except the first one, as noted) assumes that we already have a group and a subgroup. Thus, 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 selfconjugate 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 

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
Want more definitions? Check out nonstandard definitions of normal subgroup and historical definitions of normal subgroup. Many of these definitions are actually useful!
QUICK BITE: Some people talk of a normal subset of a group: a subset that is a union of conjugacy classes of elements. Then, a normal subgroup is simply a normal subset that also happens to be a subgroup. Moreover, the subgroup generated by a normal subset is a normal subgroup, though there can exist nonnormal subsets that generate normal subgroups.
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
VIEW RELATED: Analogues of this  Variations of this  Opposites of this [SHOW MORE]
This article defines a subgroup property that is pivotal (viz important) among existing subgroup properties
View a list of pivotal subgroup properties  View a complete list of subgroup properties[SHOW MORE]
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  Related subgroup property dissatisfactions
Extreme examples
 The trivial subgroup is always normal. Further information: Trivial subgroup is normal
 Every group is normal as a subgroup of itself. Further information: Every group is normal in itself
Examples
 High occurrence example: In an abelian group, every subgroup is normal (there are nonabelian groups, such as the quaternion group, where every subgroup is normal. Groups in which every subgroup is normal are called Dedekind groups, and the nonabelian ones are called Hamiltonian groups). Further information: abelian implies every subgroup is normal
 If is an internal direct product of subgroups and , both and are normal in . Further information: direct factor implies normal
 Every subgroupdefining function yields a normal subgroup (in fact, it yields a characteristic subgroup). For instance, the center, commutator subgroup and Frattini subgroup in any group are normal. Further information: subgroupdefining function value is characteristic, characteristic implies normal
Nonexamples
Here are some examples of nonnormal subgroups:
 In the symmetric group on three letters, the twoelement 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
 More generally, in any dihedral group of degree at least , the twoelement subgroup generated by a reflection is not normal. Further information: Twoelement 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 nonAbelian group is the alternating group on five letters.
Specific examples for small finite groups
Here are examples of subgroups that satisfy the property of being normal:
 Group part  Subgroup part 

A3 in S3  Symmetric group:S3  Cyclic group:Z3 
A4 in S4  Symmetric group:S4  Alternating group:A4 
Center of M16  M16  Cyclic group:Z4 
Center of central product of D8 and Z4  Central product of D8 and Z4  Cyclic group:Z4 
Center of dihedral group:D16  Dihedral group:D16  Cyclic group:Z2 
Center of dihedral group:D8  Dihedral group:D8  Cyclic group:Z2 
Center of direct product of D8 and Z2  Direct product of D8 and Z2  Klein fourgroup 
Center of nontrivial semidirect product of Z4 and Z4  Nontrivial semidirect product of Z4 and Z4  Klein fourgroup 
Center of quaternion group  Quaternion group  Cyclic group:Z2 
Center of semidihedral group:SD16  Semidihedral group:SD16  Cyclic group:Z2 
Center of special linear group:SL(2,3)  Special linear group:SL(2,3)  Cyclic group:Z2 
Center of special linear group:SL(2,5)  Special linear group:SL(2,5)  Cyclic group:Z2 
Center of unitriangular matrix group:UT(3,p)  Unitriangular matrix group:UT(3,p)  Group of prime order 
Central subgroup generated by a nonsquare in nontrivial semidirect product of Z4 and Z4  Nontrivial semidirect product of Z4 and Z4  Cyclic group:Z2 
Cyclic maximal subgroup of dihedral group:D16  Dihedral group:D16  Cyclic group:Z8 
Cyclic maximal subgroup of dihedral group:D8  Dihedral group:D8  Cyclic group:Z4 
Cyclic maximal subgroup of semidihedral group:SD16  Semidihedral group:SD16  Cyclic group:Z8 
Cyclic maximal subgroups of quaternion group  Quaternion group  Cyclic group:Z4 
D8 in D16  Dihedral group:D16  Dihedral group:D8 
D8 in SD16  Semidihedral group:SD16  Dihedral group:D8 
Derived subgroup of M16  M16  Cyclic group:Z2 
Derived subgroup of dihedral group:D16  Dihedral group:D16  Cyclic group:Z4 
Derived subgroup of nontrivial semidirect product of Z4 and Z4  Nontrivial semidirect product of Z4 and Z4  Cyclic group:Z2 
Diagonally embedded Z4 in direct product of Z8 and Z2  Direct product of Z8 and Z2  Cyclic group:Z4 
Direct product of Z4 and Z2 in M16  M16  Direct product of Z4 and Z2 
First agemo subgroup of direct product of Z4 and Z2  Direct product of Z4 and Z2  Cyclic group:Z2 
First omega subgroup of direct product of Z4 and Z2  Direct product of Z4 and Z2  Klein fourgroup 
Group of integers in group of rational numbers  Group of rational numbers  Group of integers 
Klein foursubgroup of M16  M16  Klein fourgroup 
Klein foursubgroup of alternating group:A4  Alternating group:A4  Klein fourgroup 
Klein foursubgroups of dihedral group:D8  Dihedral group:D8  Klein fourgroup 
Noncentral Z4 in M16  M16  Cyclic group:Z4 
Noncharacteristic order two subgroups of direct product of Z4 and Z2  Direct product of Z4 and Z2  Cyclic group:Z2 
Normal Klein foursubgroup of symmetric group:S4  Symmetric group:S4  Klein fourgroup 
Q8 in SD16  Semidihedral group:SD16  Quaternion group 
Q8 in central product of D8 and Z4  Central product of D8 and Z4  Quaternion group 
SL(2,3) in GL(2,3)  General linear group:GL(2,3)  Special linear group:SL(2,3) 
Subgroup generated by a noncommutator square in nontrivial semidirect product of Z4 and Z4  Nontrivial semidirect product of Z4 and Z4  Cyclic group:Z2 
Z2 in V4  Klein fourgroup  Cyclic group:Z2 
Z4 in direct product of Z4 and Z2  Direct product of Z4 and Z2  Cyclic group:Z4 
Here are examples of subgroups that do not satisfy the property of being normal:
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)
Firstorder description
This subgroup property is a firstorder subgroup property, viz., it has a firstorder description in the theory of groups.
View a complete list of firstorder subgroup properties
The subgroup property of normality can be expressed in firstorder 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 restrictionexpressible 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 restrictionexpressible subgroup properties  View the function restriction formalism chart for a graphic placement of this property
[SHOW MORE]
Function restriction expression 
is a normal subgroup of if ... 
This means that normality is ... 
Additional comments

classpreserving automorphism function 
every classpreserving automorphism of sends every element of to within 
the invariance property for classpreserving automorphisms 

classpreserving automorphism endomorphism 
every classpreserving automorphism of restricts to an endomorphism of 
the endoinvariance property for classpreserving automorphisms; i.e., it is the invariance property for classpreserving automorphism, which is a property stronger than the property of being an endomorphism 

classpreserving automorphism automorphism 
every classpreserving automorphism of restricts to an automorphism of 
the autoinvariance property for classpreserving automorphisms; i.e., it is the invariance property for classpreserving automorphism, which is a groupclosed property of automorphisms 
classpreserving automorphism to automorphism is right tight for normality

subgroupconjugating automorphism function 
every subgroupconjugating automorphism of sends every element of to within 
the invariance property for subgroupconjugating automorphisms 

subgroupconjugating automorphism endomorphism 
every subgroupconjugating automorphism of restricts to an endomorphism of 
the endoinvariance property for subgroupconjugating automorphisms; i.e., it is the invariance property for subgroupconjugating automorphism, which is a property stronger than the property of being an endomorphism 

subgroupconjugating automorphism automorphism 
every subgroupconjugating automorphism of restricts to an automorphism of 
the autoinvariance property for subgroupconjugating automorphisms; i.e., it is the invariance property for subgroupconjugating automorphism, which is a groupclosed property of automorphisms 

normal automorphism function 
every normal automorphism of sends every element of to within 
the invariance property for normal automorphisms 

normal automorphism endomorphism 
every normal automorphism of restricts to an endomorphism of 
the endoinvariance property for normal automorphisms; i.e., it is the invariance property for normal automorphism, which is a property stronger than the property of being an endomorphism 

normal automorphism automorphism 
every normal automorphism of restricts to an automorphism of 
the autoinvariance property for normal automorphisms; i.e., it is the invariance property for normal automorphism, which is a groupclosed property of automorphisms 
normal automorphism to automorphism is left tight for normality (also right tight)

weakly normal automorphism function 
every weakly normal automorphism of sends every element of to within 
the invariance property for weakly normal automorphisms 

weakly normal automorphism endomorphism 
every weakly normal automorphism of restricts to an endomorphism of 
the endoinvariance property for weakly normal automorphisms; i.e., it is the invariance property for weakly normal automorphism, which is a property stronger than the property of being an endomorphism 

monomial automorphism function 
every monomial automorphism of sends every element of to within 
the invariance property for monomial automorphisms 

monomial automorphism endomorphism 
every monomial automorphism of restricts to an endomorphism of 
the endoinvariance property for monomial automorphisms; i.e., it is the invariance property for monomial automorphism, which is a property stronger than the property of being an endomorphism 

strong monomial automorphism function 
every strong monomial automorphism of sends every element of to within 
the invariance property for strong monomial automorphisms 

strong monomial automorphism endomorphism 
every strong monomial automorphism of restricts to an endomorphism of 
the endoinvariance property for strong monomial automorphisms; i.e., it is the invariance property for strong monomial automorphism, which is a property stronger than the property of being an endomorphism 

strong monomial automorphism automorphism 
every strong monomial automorphism of restricts to an automorphism of 
the autoinvariance property for strong monomial automorphisms; i.e., it is the invariance property for strong monomial automorphism, which is a groupclosed property of automorphisms 

For more function restriction expressions for normality, check out Nonstandard definitions of normal subgroup#Definition via function restriction expression.
Relation implication expression
This subgroup property is a relation implicationexpressible subgroup property: it can be defined and viewed using a relation implication expression
View other relation implicationexpressible 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 Iautomorphisms. An Iautomorphism 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 Iautomorphisms.
 Treating the variety of groups as a variety of algebras with zero, the normal subgroups are precisely the ideals.
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) 
Centerfixing automorphisminvariant subgroup, Characteristic subgroup of direct factor, Characteristicpotentially characteristic subgroup, Cofactorial automorphisminvariant subgroup, IAautomorphisminvariant subgroup, Normalextensible automorphisminvariant subgroup, Normalpotentially characteristic subgroup, Normalpotentially relatively characteristic subgroup, Semistrongly imagepotentially characteristic subgroup, Strongly imagepotentially characteristic subgroup, Upper join of characteristic subgroupsFULL 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) 
Conjugacyclosed normal subgroup, Locally inner automorphismbalanced subgroup, Normal subgroup whose center is contained in the center of the whole group, Normal subgroup whose focal subgroup equals its derived subgroup, SCABsubgroup, Transitively normal subgroupFULL 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, Conjugacyclosed normal subgroup, Direct factor over central subgroup, Endomorphism kernel, Intermediately endomorphism kernel, Join of finitely many direct factors, Jointransitively central factor, Locally inner automorphismbalanced subgroup, Normal AEPsubgroup, Normal subgroup having a 1closed transversal, Normal subgroup in which every subgroup characteristic in the whole group is characteristic, Normal subgroup whose focal subgroup equals its derived subgroup, Poweringinvariant normal subgroup, Quotientpoweringinvariant subgroup, Rightquotienttransitively central factor, SCABsubgroup... further resultsFULL 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, Amalgamcharacteristic subgroup, Amalgamstrictly characteristic subgroup, Centerfixing automorphisminvariant subgroup, Central factor, Class two normal subgroup, Commutatorincenter subgroup, Conjugacyclosed normal subgroup, Dedekind normal subgroup, Direct factor over central subgroup, Hereditarily normal subgroup, Jointransitively central factor, Nilpotent normal subgroup, SCABsubgroup, Transitively normal subgroupFULL LIST, MORE INFO 
normal subgroup versus central subgroup 
abelian group

[SHOW MORE]
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 powerclosed characteristic subgroup, Fully invariantpotentially fully invariant subgroup, Imagepotentially fully invariant subgroup, Injective endomorphisminvariant subgroup, Normalpotentially fully invariant subgroup, Normalitypreserving endomorphisminvariant subgroup, Potentially fully invariant subgroup, Retractioninvariant characteristic subgroup, Retractioninvariant normal subgroup, Strictly characteristic subgroupFULL LIST, MORE INFO 
group in which every normal subgroup is fully invariant

strictly characteristic subgroup 
invariant under surjective endomorphisms 
(via characteristic) 
(via characteristic) (see also list of examples) 
Characteristic subgroupFULL LIST, MORE INFO 

isomorphfree subgroup 
no other isomorphic subgroup 
(via characteristic) 
(via characteristic) (see also list of examples) 
Characteristic subgroup, Hallrelatively weakly closed subgroup, Injective endomorphisminvariant subgroup, Intermediately characteristic subgroup, Isomorphautomorphic normal subgroup, Isomorphcontaining subgroup, Isomorphnormal characteristic subgroup, Isomorphnormal subgroup, Subisomorphfree subgroupFULL LIST, MORE INFO 
group in which every normal subgroup is isomorphfree

isomorphcontaining subgroup 
contains all isomorphic subgroups 
(via characteristic) 
(via characteristic) (see also list of examples) 
Characteristic subgroup, Injective endomorphisminvariant subgroupFULL LIST, MORE INFO 

injective endomorphisminvariant subgroup 
invariant under injective endomorphisms 
(via characteristic) 
(via characteristic) (see also list of examples) 
Characteristic subgroupFULL LIST, MORE INFO 

transitively normal subgroup 
any normal subgroup is normal in whole group 

(see also list of examples) 
FULL LIST, MORE INFO 
Tgroup

cocentral subgroup 
product with center is whole group 
cocentral implies normal 
normal not implies cocentral 
Central factor, Conjugacyclosed normal subgroup, Rightquotienttransitively central factor, SCABsubgroup, Transitively normal subgroupFULL LIST, MORE INFO 
abelian group

SCABsubgroup 
subgroupconjugating automorphism restricts to subgroupconjugating automorphism of subgroup 


Transitively normal subgroupFULL LIST, MORE INFO 

conjugacyclosed normal subgroup 
conjugacyclosed 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 subgroupFULL LIST, MORE INFO 

abelianquotient subgroup 
contains the derived subgroup 


Nilpotentquotient subgroupFULL 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 1closed transversal, Poweringinvariant normal subgroup, Quotientpoweringinvariant subgroupFULL LIST, MORE INFO

endomorphism kernel 
occurs as the kernel of an endomorphism 
(by definition) 
normal not implies endomorphism kernel 
Poweringinvariant normal subgroup, Quotientpoweringinvariant subgroupFULL 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  quotienttransitive intersectionclosed joinclosed  trim  inverse image condition  image condition  centralizerclosed 
STRONGER PROPERTIES DISSATISFYING SPECIFIC METAPROPERTIES: transitive  intermediate subgroup condition  transfer condition  quotienttransitive intersectionclosed joinclosed  trim  inverse image condition  image condition  centralizerclosed 
Conjunction with other properties
Important conjunctions of normality with other subgroup properties are in the table below:
[SHOW MORE]
Conjunction 
Other component of conjunction 
Intermediate notions 
Additional comments

conjugacyclosed normal subgroup 
conjugacyclosed 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 subgroupFULL LIST, MORE INFO 

transitively normal subgroup 
CEPsubgroup 
FULL LIST, MORE INFO 

normal Sylow subgroup 
Sylow subgroup 
Complemented fully invariant subgroup, Complemented homomorphcontaining subgroup, Complemented normal subgroup, Fully invariant subgroup, Homomorphcontaining subgroup, Isomorphfree 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, Normalhomomorphcontaining subgroup, Ordernormal subgroup, Orderunique subgroup, Subhomomorphcontaining subgroupFULL LIST, MORE INFO 

normal Hall subgroup 
Hall subgroup 
Complemented fully invariant subgroup, Complemented homomorphcontaining subgroup, Complemented normal subgroup, Fully invariant subgroup, Homomorphcontaining subgroup, Isomorphfree 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, Normalhomomorphcontaining subgroup, Ordernormal subgroup, Orderunique subgroup, Subhomomorphcontaining subgroupFULL 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:
[SHOW MORE]
Conjunction 
Other component of conjunction 
Intermediate notions 
Additional comments

abelian normal subgroup 
abelian group 
Class two normal subgroup, Commutatorincenter subgroup, Dedekind normal subgroup, Nilpotent normal subgroup, Normal subgroup whose inner automorphism group is central in automorphism group, Solvable normal subgroupFULL LIST, MORE INFO 

cyclic normal subgroup 
cyclic group 
Abelian normal subgroup, Commutatorincenter 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 subgroupFULL LIST, MORE INFO 

solvable normal subgroup 
solvable group 
FULL LIST, MORE INFO 

nilpotent normal subgroup 
nilpotent group 
Solvable normal subgroupFULL LIST, MORE INFO 

simple normal subgroup 
simple group 
FULL LIST, MORE INFO 

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 subgroupsFULL LIST, MORE INFO 

finite normal subgroup 
finite group 
Amalgamcharacteristic subgroup, Amalgamnormalsubhomomorphcontaining subgroup, Amalgamstrictly characteristic subgroup, Finitely generated normal subgroup, Join of finite normal subgroups, Normal closure of finite subset, Periodic normal subgroup, Potentially normalsubhomomorphcontaining subgroup, Poweringinvariant normal subgroup, Quotientpoweringinvariant subgroupFULL LIST, MORE INFO 

periodic normal subgroup 
periodic group 
Amalgamcharacteristic subgroup, Amalgamnormalsubhomomorphcontaining subgroup, Amalgamstrictly characteristic subgroup, Potentially normalsubhomomorphcontaining subgroupFULL LIST, MORE INFO 

finitely generated normal subgroup 
finitely generated group 
Normal closure of finite subsetFULL 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:
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) 
2hypernormalized subgroup, 2subnormal subgroup, 3subnormal subgroup, 4subnormal subgroup, Asymptotically fixeddepth jointransitively subnormal subgroup, Central factor of normal subgroup, Direct factor of normal subgroup, Finitarily hypernormalized subgroup, Finiteconjugatejoinclosed subnormal subgroup, Intermediately jointransitively subnormal subgroup, Join of finitely many 2subnormal subgroups, Jointransitively 2subnormal subgroup, Jointransitively subnormal subgroup, Linearbound jointransitively subnormal subgroup, Modular 2subnormal subgroup, Modular subnormal subgroup, Normal subgroup of characteristic subgroup, Permutable 2subnormal subgroup, Permutable subnormal subgroup, Subnormalpermutable subnormal subgroupFULL LIST, MORE INFO 
contrasting subnormality of various depths 
Tgroup

permutable subgroup (also called quasinormal subgroup) 
permutes with every subgroup 
normal implies permutable 
permutable not implies normal (see also list of examples) 
Permutable 2subnormal subgroupFULL LIST, MORE INFO 

group in which every permutable subgroup is normal

2subnormal subgroup 
normal subgroup of normal subgroup 

normality is not transitive (see also list of examples) 
2hypernormalized subgroup, Central factor of normal subgroup, Direct factor of normal subgroup, Jointransitively 2subnormal subgroup, Modular 2subnormal subgroup, Normal subgroup of characteristic subgroup, Permutable 2subnormal subgroup, Transitively normal subgroup of normal subgroupFULL LIST, MORE INFO 
contrasting subnormality of various depths 
Tgroup

[SHOW MORE]
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions 
Comparison 
Collapse

3subnormal subgroup 
normal of normal of normal 


2subnormal subgroup, Normal subgroup of characteristic subgroupFULL LIST, MORE INFO 
contrasting subnormality of various depths 
Tgroup

4subnormal subgroup 
2subnormal of 2subnormal 


2subnormal subgroup, Normal subgroup of characteristic subgroupFULL LIST, MORE INFO 
contrasting subnormality of various depths 
Tgroup

ascendant subgroup 



Hypernormalized subgroup, Permutable subgroup, Subnormal subgroupFULL LIST, MORE INFO 


descendant subgroup 



Conjugatepermutable subgroup, Intersection of subnormal subgroups, Subnormal subgroupFULL LIST, MORE INFO 


serial subgroup 



Ascendant subgroup, Hypernormalized subgroup, Subnormal subgroupFULL LIST, MORE INFO 


hypernormalized subgroup 
iterated normalizers reach whole group 

(see also list of examples) 
2hypernormalized subgroup, Finitarily hypernormalized subgroupFULL LIST, MORE INFO 


finitarily hypernormalized subgroup 
iterated normalizers reach whole group in finite number of steps 


2hypernormalized subgroupFULL LIST, MORE INFO 


2hypernormalized subgroup 
normalizer is normal 


FULL LIST, MORE INFO 


pronormal subgroup 
conjugate to any conjugate in their join 
normal implies pronormal 
pronormal not implies normal (see also list of examples) 
Jointransitively pronormal subgroup, Subgroup whose join with any distinct conjugate is the whole group, Transferclosed pronormal subgroupFULL LIST, MORE INFO 
subnormaltonormal and normaltocharacteristic 
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 groupFULL LIST, MORE INFO 
(above) 

paranormal subgroup 

(via pronormal) 

Pronormal subgroupFULL LIST, MORE INFO 
(above) 

weakly normal subgroup 

(via pronormal) 

NEsubgroup, Paranormal subgroupFULL LIST, MORE INFO 
(above) 

intermediately subnormaltonormal subgroup 

(via pronormal) 
(see also list of examples) 
NEsubgroup, Paranormal subgroup, Polynormal subgroup, Pronormal subgroup, Weakly normal subgroup, Weakly pronormal subgroupFULL LIST, MORE INFO 
(above) 

conjugatepermutable subgroup 
permutes with its conjugate subgroups 


2hypernormalized subgroup, 2subnormal subgroup, Permutable 2subnormal subgroup, Permutable subgroup, Permutable subgroup of normal subgroupFULL LIST, MORE INFO 


modular subgroup 



Modular 2subnormal subgroup, Modular subnormal subgroup, Permutable subgroupFULL LIST, MORE INFO 


automorphpermutable subgroup 



Normal subgroup of characteristic subgroup, Permutable subgroupFULL LIST, MORE INFO 


elliptic subgroup 



Permutable subgroupFULL 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  quotienttransitive intersectionclosed joinclosed  trim  inverse image condition  image condition  centralizerclosed 
WEAKER PROPERTIES DISSATISFYING SPECIFIC METAPROPERTIES: transitive  intermediate subgroup condition  transfer condition  quotienttransitive intersectionclosed joinclosed  trim  inverse image condition  image condition  centralizerclosed 
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.
Further information: Category: Subgroup operators related to normality
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.

idealdetermined variety of algebras (universal algebra) 
ideal in that variety 
An idealdetermined 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 idealdetermined, with the ideals being the normal subgroups.

variety of algebras (universal algebra) 
Iautomorphisminvariant subalgebra 
invariant under Iautomorphisms, which are the automorphism described by formulas that universally give automorphisms. 
inner automorphisms are Iautomorphisms 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 
leftnormal subsemigroup, rightnormal subsemigroup 


Facts
Isomorphism theorems
Theorem name/number 
Statement

First isomorphism theorem 
If is a surjective homomorphism, then the kernel of 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 such that is contained in the normalizer of , then is normal in and .

Third isomorphism theorem 
If with both normal in , then is normal in and .

Fourth isomorphism theorem (lattice isomorphism theorem) 
If is normal in , 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

Transitive subgroup property 
No 
Normality is not transitive 
Can have , normal in , is normal in , 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 intersectionclosed subgroup property 
Yes 
Normality is strongly intersectionclosed 
Given , all normal in , so is

Strongly joinclosed subgroup property 
Yes 
Normality is strongly joinclosed 
Given , all normal in , so is

Quotienttransitive subgroup property 
Yes 
Normality is quotienttransitive 
, normal in , normal in , then normal in

Intermediate subgroup condition 
Yes 
Normality satisfies intermediate subgroup condition 
, normal in , then normal in

Transfer condition 
Yes 
Normality satisfies transfer condition 
, normal in , then normal in

Image condition 
Yes 
Normality satisfies image condition 
normal in , surjective, then normal in

Inverse image condition 
Yes 
Normality satisfies inverse image condition 
normal in , homomorphism, then is normal in

Upper joinclosed subgroup property 
Yes 
Normality is upper joinclosed 
, intermediate subgroups, normal in each, then is normal in their join

Commutatorclosed subgroup property 
Yes 
Normality is commutatorclosed 
both normal, then normal in

Centralizerclosed subgroup property 
Yes 
Normality is centralizerclosed 
normal in , then also normal in

Direct productclosed subgroup property 
Yes 
Normality is direct productclosed 
normal in implies direct product of s normal in direct product of s.

For more information on these metaproperties:
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Tautology when whole group is abelian
This subgroup property is an abeliantautological subgroup property: it is always true for a subgroup of an abelian group.
View a complete list of abeliantautological subgroup properties
Every subgroup of an abelian group is normal. A group in which every subgroup is normal is termed a Dedekind group, so this says that every abelian group is Dedekind. There do exist nonabelian Dedekind groups, such as the quaternion group.
For full proof, refer: Abelian implies every subgroup is normal
Further information: Dedekind not implies abelian
Transitivity
NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitiveView variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitiveView facts related to transitivity of subgroup properties  View a survey article on disproving transitivity
Normality is not transitive. That is, it is possible to have groups such that is normal in and is normal in but is not normal in . The smallest counterexample is where is the dihedral group of order eight, is a nonnormal subgroup of order two and is a Klein foursubgroup containing it.
For full proof, refer: Normality is not transitive
Further information: there exist subgroups of arbitrarily large subnormal depth, Characteristic of normal implies normal, Characteristicity is transitive, Left transiter of normal is characteristic
Trimness
This subgroup property is trim  it is both trivially true (true for the trivial subgroup) and identitytrue (true for a group as a subgroup of itself).
View other trim subgroup properties  View other trivially true subgroup properties  View other identitytrue subgroup properties
Normality is a trim subgroup property: both the trivial subgroup and the improper subgroup (i.e., the whole group itself) are normal as subgroups of the whole group.
For full proof, refer: Every group is normal in itself, trivial subgroup is normal
Further information: Invariance implies identitytrue, Endoinvariance implies trim
Intersectionclosedness
YES: This subgroup property is intersectionclosed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
In fact, since the property is also true for every group as a subgroup of itself, it is a strongly intersectionclosed subgroup property.
ABOUT THIS PROPERTY: View variations of this property that are intersectionclosed  View variations of this property that are not intersectionclosed
ABOUT INTERSECTIONCLOSEDNESS: View all intersectionclosed subgroup properties (or, strongly intersectionclosed properties)  View all subgroup properties that are not intersectionclosed  Read a survey article on proving intersectionclosedness  Read a survey article on disproving intersectionclosedness
On account of normality being an invariance property, it is strongly intersectionclosed, viz., an arbitrary (possibly empty) intersection of normal subgroups is again normal in the whole group. (Note that the empty intersection is by default the whole group). In other words, if is a collection of normal subgroups of , then is normal in .
For full proof, refer: Normality is strongly intersectionclosed
Further information: Invariance implies strongly intersectionclosed, Characteristicity is strongly intersectionclosed, Full invariance is strongly intersectionclosed
Joinclosedness
YES: This subgroup property is joinclosed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
In fact, since the property is also true for the trivial subgroup in any group, it is a strongly joinclosed subgroup property.
ABOUT THIS PROPERTY: View variations of this property that are joinclosed  View variations of this property that are not joinclosed
ABOUT JOINCLOSEDNESS: View all joinclosed subgroup properties (or, strongly joinclosed properties)  View all subgroup properties that are not joinclosed  Read a survey article on proving joinclosedness  Read a survey article on disproving joinclosedness
Since normality is an invariance property with respect to functions that are all endomorphisms, it is also strongly joinclosed, viz., the subgroup generated by an arbitrary family of normal subgroups is again normal. (Note that the empty join is by default the trivial subgroup). In other words, if is a collection of normal subgroups of , then the join is normal in .
For full proof, refer: Normality is strongly joinclosed
Further information: Endoinvariance implies strongly joinclosed, Characteristicity is strongly joinclosed, Full invariance is strongly joinclosed
Intermediate subgroup condition
YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition  View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition  View facts about intermediate subgroup condition
If is a normal subgroup of and is a subgroup of containing , then is normal in . We code this fact by saying that normality satisfies the intermediate subgroup condition.
The essential reason for this is that normality can be expressed in the function restriction formalism as a leftinner subgroup property.
For full proof, refer: Normality satisfies intermediate subgroup condition
Further information: Leftinner implies intermediate subgroup condition, Leftextensibilitystable implies intermediate subgroup condition, Central factor satisfies intermediate subgroup condition, Direct factor satisfies intermediate subgroup condition
Transfer condition
YES: This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
View other subgroup properties satisfying the transfer condition
Normality satisfies the transfer condition. In other words, if is normal, and is any subgroup of then is a normal subgroup of .
For full proof, refer: Normality satisfies transfer condition
Inverse image condition
This subgroup property satisfies the inverse image condition. In other words, the inverse image under any homomorphism of a subgroup satisfying the property also satisfies the property. In particular, this property satisfies the transfer condition and intermediate subgroup condition.
Normality satisfies the inverse image condition. That is, if is a homomorphism and is a normal subgroup of , then is a normal subgroup of .
For full proof, refer: Normality satisfies inverse image condition
Quotienttransitivity
This subgroup property is quotienttransitive: the corresponding quotient property is transitive.
View a complete list of quotienttransitive subgroup properties
The property of normality is a quotienttransitive subgroup property. That is, if are groups such that is normal in and is normal in , then is normal in .
For full proof, refer: Normality is quotienttransitive
Further information: Quotientbalanced implies quotienttransitive, Characteristicity is quotienttransitive, Full invariance is quotienttransitive
Image condition
YES: This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property
View other subgroup properties satisfying image condition
Under any surjective homomorphism, the image of a normal subgroup is normal. In other words, if is a normal subgroup of and is a surjective homomorphism, then is normal in .
For full proof, refer: Normality satisfies image condition
Further information: Subnormality satisfies image condition, Pronormality satisfies image condition, Characteristicity does not satisfy image condition
Centralizerclosedness
This subgroup property is centralizerclosed: the centralizer of any subgroup with this property, in the whole group, again has this property
View other centralizerclosed subgroup properties
The centralizer of a normal subgroup is always normal. Thus, if is normal in , so is . This follows from the general fact that normality can be described as an autoinvariance property: an invariance property with respect to a property of automorphisms (namely, inner automorphisms).
For full proof, refer: Normality is centralizerclosed
Further information: Autoinvariance implies centralizerclosed, Characteristicity is centralizerclosed, Automorphconjugacy is centralizerclosed
Commutatorclosedness
This subgroup property is commutatorclosed: the commutator of two subgroups each with the property, also has the property.
View other commutatorclosed subgroup properties
A commutator of two normal subgroups is normal. Thus, if are normal subgroups of , so is . However, it is not necessarily true that the commutator of a normal subgroup and an arbitrary subgroup be normal. It is true, though, that the commutator of the whole group and any subgroup (in fact, any subset) is normal, and also that the commutator of two subgroups is normal in their join.
For full proof, refer: Normality is commutatorclosed
Further information: Commutator of a normal subgroup and a subgroup not implies normal, Commutator of the whole group and a subset implies normal, Commutator of two subgroups is normal in join, Endoinvariance implies commutatorclosed, Characteristicity is commutatorclosed, Full invariance is commutatorclosed
Upper joinclosedness
This subgroup property is upper joinclosed, viz., if a subgroup has the property in a collection of intermediate subgroups, it also has the property in their join
View other such properties
If is a subgroup of , and are intermediate subgroups such that for all , then .
In fact, there is a unique largest subgroup of inside which is normal. This subgroup is termed the normalizer of in .
For full proof, refer: Normality is upper joinclosed
Further information: Characteristicity is not upper joinclosed, 2subnormality is not upper joinclosed
Direct productclosedness
This subgroup property is direct productclosed: it is closed under taking arbitrary direct products of groups
If is a nonempty indexing set and is normal in for each , then the direct product of the s is normal in the direct product of the s.
For full proof, refer: Normality is direct productclosed
Arguesianness
Normality is an Arguesian subgroup property. In other words, the collection of normal subgroups of a group form an Arguesian lattice (the fact that they form a lattice follows from the fact that normality is trim, joinclosed and intersectionclosed).
For full proof, refer: Normality is Arguesian
Effect of property operators
For more information on these property operators:
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The left transiter
Applying the left transiter to this property gives: characteristic subgroup
The left transiter of normality is the property of being characteristic. In other words, if is a subgroup of such that whenever is normal in some bigger group so is , then is characteristic in . Conversely, if is characteristic in and is normal in , is normal in <mah>K</math>.
Characteristicity is the balanced subgroup property corresponding to automorphisms. This is a consequence of the fact that every group can be embedded as a normal fully normalized subgroup in another group. For full proof, refer: Left transiter of normal is characteristic
Further information: Characteristic of normal implies normal, normality is not transitive, characteristicity is transitive, normal upperhook fully normalized implies characteristic
The right transiter
Applying the right transiter to this property gives: transitively normal subgroup
The right transiter of normality is the property of being transitively normal. This is the balanced subgroup property corresponding to normal automorphisms.
The subordination operator
Applying the subordination operator to this property gives: subnormal subgroup
The hereditarily operator
Applying the hereditarily operator to this property gives: hereditarily normal subgroup
The result of applying the hereditarily operator to the subgroup property of being normal gives the subgroup property of being hereditarily normal: viz., a subgroup of a group is termed hereditarily normal if every subgroup of is normal in . Note that being hereditarily normal is equivalent to being transitively normal and being a Dedekind group: a group in which every subgroup is normal.
The center is an example of a hereditarily normal subgroup.
The upwardclosure operator
Applying the upwardclosure operator to this property gives: upwardclosed normal subgroup
The result of applying the upward closure operator to the subgroup property of being normal gives the property of being upwardclosed normal, viz., a subgroup is upwardclosed normal in if for any intermediate subgroup of , is normal in .
The commutator subgroup is an example of an upwardclosed normal subgroup.
The maximal proper operator
Applying the maximal proper operator to this property gives: maximal normal subgroup
A maximal normal subgroup is a proper normal subgroup not contained in any other proper normal subgroup. A subgroup is maximal normal if and only if it is normal and the quotient is a simple group.
The minimal operator
Applying the minimal operator to this property gives: minimal normal subgroup
A minimal normal subgroup is a nontrivial normal subgroup that does not contain any other nontrivial normal subgroup.
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 builtin 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 builtin GAP commandView subgroup properties for which all subgroups can be listed with builtin 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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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 welldefined.
Normal subgroups were earlier termed invariant subgroups (because they were invariant under inner automorphisms) and also termed selfconjugate subgroups (because a normal subgroup is precisely a subgroup that equals every conjugate).
Origin of the term
This term was 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.
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, 10digit ISBN 0471433349, 13digit ISBN 9780471433347^{More info} 
82 

formal definition,, and Theorem 6 giving equivalent formulations. Also, Page 80 (first use). 

Topics in Algebra by I. N. Herstein^{More info} 
50 
Section 2.6 
formal definition 

Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189^{More info} 
41 
Section 1.5 
definition introduced through proposition 
Google Books

Algebra by Serge Lang, ISBN 038795385X^{More info} 
14 

definition in paragraph 
Google Books

Algebra by Michael Artin, ISBN 0130047635, 13digit ISBN 9780130047632^{More info} 
52 
Point (4.8) 
formal definition, followed by equivalent definitioncumproposition in (4.9) 

Basic Algebra: Groups, Rings, and Fields by Paul Moritz Cohn, ISBN 1852335874, 13digit ISBN 9781852335878^{More 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, 13digit ISBN 9780070041240^{More 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 0387945261^{More info} 
6 

definition in paragraph 
Google Books

A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More 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, 13digit ISBN 9780387942858^{More info} 
30 

formal definition 
Google Books

Online lecture notes
External links
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Definition links
Names in other languages:German: Normalteiler; French: Sousgroupe normal; Spanish: Subgrupo normal; Italian: Sottogruppo normale
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