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 $H$ of a group $G$ is normal in $G$ if ...	Applications to...	Additional comments
1	Homomorphism kernel	it is the kernel of a homomorphism from the group.	there is a homomorphism $\varphi$ from $G$ to a group $K$ such that the kernel of $\varphi$ is precisely $H\!$. In other words, $\varphi(x) \!$ is the identity element of $K$ if and only if $x \in H$.	proving normality	In this case, we do not need to separately check that $H$ 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 $g \in G$, $gHg^{-1} \subseteq H$. More explicitly, for all $g \in G, h \in H$, we have $ghg^{-1} \in H$.	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 $g$ in $G$, $gHg^{-1} = H$.		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 $g$ in $G$, $gH = Hg$.	proving normality	When we say $gH = Hg$, we only mean equality as sets. It is not necessary that $gh = hg$ for $h \in H$. That stronger condition defines central subgroup.
5	Union of conjugacy classes	it is a union of conjugacy classes.	$H$ is a union of conjugacy classes in $G$
6	Commutator inside	it contains its commutator with the whole group.	the commutator $[H,G]$ (which coincides with the commutator $[G,H]$) is contained in $H$.	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 $A$ for $G$ and $B$ for $H$, we have $aba^{-1} \in H$ and $a^{-1}ba \in H$ for all $a \in A, b\in B$.	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 $H \!$ of a group $G \!$, we denote the normality of $\! H$ in $\! G$ by $H \underline{\triangleleft} G$ or $G \underline{\triangleright} H$^Notations. In words, we say that $\! H$ is normal in $\! G$ or a normal subgroup of $\! G$.

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 $g$ is conjugation by $g^{-1}$)	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.
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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 | Related subgroup property dissatisfactions

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 $G$ is an internal direct product of subgroups $H$ and $K$, both $H$ and $K$ are normal in $G$. 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 $3$, 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
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Dihedral group:D8

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 $\varphi:G \to H$ is a surjective homomorphism, and $N$ is the kernel of $\varphi$, then $N$ is a normal subgroup, and if $\alpha:G \to G/N$ is the quotient map, then there is a unique isomorphism $\psi:G/N \to H$ such that $\psi \circ \alpha = \varphi$.
Second isomorphism theorem (diamond isomorphism theorem)	If $N,H \le G$ are subgroups such that $H$ is contained in the normalizer of $N$, then $N$ is normal in $NH$ and $NH/N \cong H/(H \cap N)$.
Third isomorphism theorem	If $H \le K \le G$ are groups with both $H,K$ normal in $G$, then $H$ is normal in $K$ and $(G/H)/(K/H) \cong G/K$.
Fourth isomorphism theorem (lattice isomorphism theorem)	If $H$ is normal in $G$, then there is a bijective correspondence between subgroups of $G/H$ and subgroups of $G$ containing $H$, 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 $H \le G$ and $G$ is abelian, then $H$ is normal in $G$.
transitive subgroup property	No	Normality is not transitive	We can have $H \le K \le G$ such that $H$ is normal in $K$ and $K$ is normal in $G$ but $H$ is not normal in $G$.
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 $H_i, i \in I$ are all normal subgroups of a group $G$, then the intersection of subgroups $\bigcap_{i \in I} H_i$ is also a normal subgroup of $G$.
strongly join-closed subgroup property	Yes	Normality is strongly join-closed	If $H_i, i \in I$ are all normal subgroups of $G$, then the join of subgroups $\langle H_i \rangle_{i \in I}$ is also a normal subgroup of $G$.
quotient-transitive subgroup property	Yes	Normality is quotient-transitive	If $H \le K \le G$ are groups such that $H$ is normal in $G$ and $K/H$ is normal in $G/H$, then $K$ is normal in $G$
intermediate subgroup condition	Yes	Normality satisfies intermediate subgroup condition	If $H \le K \le G$ are groups such that $H$ is normal in $G$, then $H$ is normal in $K$.
transfer condition	Yes	Normality satisfies transfer condition	If $H, K \le G$ are groups such that $H$ is normal in $G$, then we must have that $H \cap K$ normal in $K$
image condition	Yes	Normality satisfies image condition	If $H$ is a normal subgroup of $G$ and $\varphi:G \to K$ is a surjective homomorphism of groups, then $\varphi(H)$ normal in $K$
inverse image condition	Yes	Normality satisfies inverse image condition	If $H$ is a normal subgroup of $G$ and $\varphi:K \to G$ is a homomorphism of groups, then $\varphi^{-1}(H)$ is normal in $K$.
upper join-closed subgroup property	Yes	Normality is upper join-closed	If $H \le G$, and $K_i, i \in I$ are all subgroups of $G$ containing $H$ such that $H$ normal in each $K_i$, then we must have that $H$ is normal in the join of subgroups $\langle K_i \rangle_{i \in I}$.
commutator-closed subgroup property	Yes	Normality is commutator-closed	If $H, K$ are both normal subgroups of a group $G$, then the commutator $[H,K]$ normal in $G$
centralizer-closed subgroup property	Yes	Normality is centralizer-closed	If $H$ is a normal subgroup of $G$, the centralizer $C_G(H)$ is also normal in $G$.
direct product-closed subgroup property	Yes	Normality is direct product-closed	If $I$ is an indexing set, $G_i, i \in I$ are groups, and $H_i, i \in I$ are such that each $H_i$ is normal in the corresponding $G_i$, then in the external direct product of the $G_i$s, the subgroup given by the external direct product of the $H_i$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 $I$ is an indexing set, $G$ is a group, and we have groups $H_i \le K_i \le G$ for each $i \in I$, such that $H_i$ is normal in $K_i$, then $\bigcap_{i \in I} H_i$ is a normal subgroup of $\bigcap_{i \in I} K_i$.
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 $H$ is a normal subgroup of a group $G$. Then, each lower central series member $\gamma_k(H)$ is a normal subgroup inside the corresponding lower central series member $\gamma_k(G)$.
partition difference condition	Yes	Normality satisfies partition difference condition	Suppose $H$ is a subgroup of a group $G$, and $H$ has a partition as a union of $H_i, i \in I$. If all except possibly one of the $H_i$s are normal subgroups of $G$, then all the $H_i$s are normal subgroups of $G$.
conditionally lattice-determined subgroup property	No	No subgroup property between normal Sylow and subnormal or between Sylow retract and retract is conditionally lattice-determined	It is possible to have a group $G$, an automorphism $\varphi$ of the lattice of subgroups, and a normal subgroup $H$ of $G$ such that $\varphi(H)$ is not normal.

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:

• Variations of normal subgroup
• Opposites of normal subgroup
• Analogues of normal subgroup

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

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

Stronger properties

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

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) (see also list of examples)	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)	|FULL LIST, MORE INFO	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 | inverse image condition | image condition | centralizer-closed |
STRONGER PROPERTIES DISSATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | trim | 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	|FULL LIST, MORE INFO
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	|FULL LIST, MORE INFO
nilpotent normal subgroup	nilpotent group	Solvable normal subgroup|FULL 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 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	|FULL LIST, MORE INFO
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 (see also list of examples)	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		(see also list of examples)	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			|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)	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 | centralizer-closed |
WEAKER PROPERTIES DISSATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | trim | inverse image condition | 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:

• Normal core of normalizer
• Hypernormalizer

Closely related to normal closure is the normal subgroup generated by a subset, which is defined as the smallest normal subgroup containing the subset, and is the normal closure of the subgroup generated by 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: $H$ is normal in $G$ if and only if:

$\forall g \in G, h \in H: \ ghg^{-1} \in H$

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	$H$ is a normal subgroup of $G$ if ...	This means that normality is ...	Additional comments
inner automorphism $\to$ function	every inner automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for inner automorphisms
inner automorphism $\to$ endomorphism	every inner automorphism of $G$ restricts to an endomorphism of $H$	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 $\to$ automorphism	every inner automorphism of $G$ restricts to an automorphism of $H$	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

[SHOW MORE]

Function restriction expression	$H$ is a normal subgroup of $G$ if ...	This means that normality is ...	Additional comments
class-preserving automorphism $\to$ function	every class-preserving automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for class-preserving automorphisms
class-preserving automorphism $\to$ endomorphism	every class-preserving automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance property for class-preserving automorphisms; i.e., it is the invariance property for class-preserving automorphism, which is a property stronger than the property of being an endomorphism
class-preserving automorphism $\to$ automorphism	every class-preserving automorphism of $G$ restricts to an automorphism of $H$	the auto-invariance property for class-preserving automorphisms; i.e., it is the invariance property for class-preserving automorphism, which is a group-closed property of automorphisms	class-preserving automorphism to automorphism is right tight for normality
subgroup-conjugating automorphism $\to$ function	every subgroup-conjugating automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for subgroup-conjugating automorphisms
subgroup-conjugating automorphism $\to$ endomorphism	every subgroup-conjugating automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance property for subgroup-conjugating automorphisms; i.e., it is the invariance property for subgroup-conjugating automorphism, which is a property stronger than the property of being an endomorphism
subgroup-conjugating automorphism $\to$ automorphism	every subgroup-conjugating automorphism of $G$ restricts to an automorphism of $H$	the auto-invariance property for subgroup-conjugating automorphisms; i.e., it is the invariance property for subgroup-conjugating automorphism, which is a group-closed property of automorphisms
normal automorphism $\to$ function	every normal automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for normal automorphisms
normal automorphism $\to$ endomorphism	every normal automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance 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 $\to$ automorphism	every normal automorphism of $G$ restricts to an automorphism of $H$	the auto-invariance property for normal automorphisms; i.e., it is the invariance property for normal automorphism, which is a group-closed property of automorphisms	normal automorphism to automorphism is left tight for normality (also right tight)
weakly normal automorphism $\to$ function	every weakly normal automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for weakly normal automorphisms
weakly normal automorphism $\to$ endomorphism	every weakly normal automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance 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 $\to$ function	every monomial automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for monomial automorphisms
monomial automorphism $\to$ endomorphism	every monomial automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance 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 $\to$ function	every strong monomial automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for strong monomial automorphisms
strong monomial automorphism $\to$ endomorphism	every strong monomial automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance 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 $\to$ automorphism	every strong monomial automorphism of $G$ restricts to an automorphism of $H$	the auto-invariance property for strong monomial automorphisms; i.e., it is the invariance property for strong monomial automorphism, which is a group-closed 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 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 $\implies$ 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, and by the inverse of the generator, 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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Origin of the concept

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

Normal subgroups were earlier termed invariant subgroups (because they were invariant under inner automorphisms) and also termed self-conjugate 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, 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

• J.S. Milne's course notes, Section 1.7, Page 15 (both the A4 and the letter versions)
• Lecture notes for Benedict Gross's lecture 4, Page 2-3 and more. You can also access the video lectures and other material for the course through the course main page.
• Lecture notes on basic group theory by James Miller, part of the solitaryroad.com website