# Normal subgroup

## 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

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 $H$ 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 $N$ 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
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
(1) and (2) Normal subgroup equals kernel of homomorphism. See also 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 $g^{-1}$). Further information: 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. Further information: 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. Further information: manipulating equations in groups
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 non-normal subsets that generate normal subgroups.

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

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, commutator 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 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.

### Specific examples for small finite groups

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

Group partSubgroup part
A3 in S3Symmetric group:S3Cyclic group:Z3
A4 in S4Symmetric group:S4Alternating group:A4
Center of M16M16Cyclic group:Z4
Center of central product of D8 and Z4Central product of D8 and Z4Cyclic group:Z4
Center of dihedral group:D16Dihedral group:D16Cyclic group:Z2
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-group
Center of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Klein four-group
Center of quaternion groupQuaternion groupCyclic group:Z2
Center of semidihedral group:SD16Semidihedral group:SD16Cyclic 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 non-square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2
Cyclic maximal subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z8
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4
Cyclic maximal subgroup of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z8
Cyclic maximal subgroups of quaternion groupQuaternion groupCyclic group:Z4
D8 in D16Dihedral group:D16Dihedral group:D8
D8 in SD16Semidihedral group:SD16Dihedral group:D8
Derived subgroup of M16M16Cyclic group:Z2
Derived subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z4
Derived subgroup of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2
Diagonally embedded Z4 in direct product of Z8 and Z2Direct product of Z8 and Z2Cyclic group:Z4
Direct product of Z4 and Z2 in M16M16Direct product of Z4 and Z2
First agemo subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2
First omega subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Klein four-group
Group of integers in group of rational numbersGroup of rational numbersGroup of integers
Klein four-subgroup of M16M16Klein four-group
Klein four-subgroup of alternating group:A4Alternating group:A4Klein four-group
Klein four-subgroups of dihedral group:D8Dihedral group:D8Klein four-group
Non-central Z4 in M16M16Cyclic group:Z4
Non-characteristic order two subgroups of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2
Normal Klein four-subgroup of symmetric group:S4Symmetric group:S4Klein four-group
Q8 in SD16Semidihedral group:SD16Quaternion group
Q8 in central product of D8 and Z4Central product of D8 and Z4Quaternion group
SL(2,3) in GL(2,3)General linear group:GL(2,3)Special linear group:SL(2,3)
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2
Z2 in V4Klein four-groupCyclic group:Z2
Z4 in direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z4

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

Group partSubgroup part
2-Sylow subgroup of general linear group:GL(2,3)General linear group:GL(2,3)Semidihedral group:SD16
A3 in A4Alternating group:A4Cyclic group:Z3
A3 in A5Alternating group:A5Cyclic group:Z3
A3 in S4Symmetric group:S4Cyclic group:Z3
A4 in A5Alternating group:A5Alternating group:A4
D8 in A6Alternating group:A6Dihedral group:D8
D8 in S4Symmetric group:S4Dihedral group:D8
Klein four-subgroup of alternating group:A5Alternating group:A5Klein four-group
Non-normal Klein four-subgroups of symmetric group:S4Symmetric group:S4Klein four-group
Non-normal subgroups of M16M16Cyclic group:Z2
Non-normal subgroups of dihedral group:D8Dihedral group:D8Cyclic group:Z2
S2 in S3Symmetric group:S3Cyclic group:Z2
S2 in S4Symmetric group:S4Cyclic group:Z2
Subgroup generated by double transposition in symmetric group:S4Symmetric group:S4Cyclic group:Z2
Twisted S3 in A5Alternating group:A5Symmetric group:S3

## 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

### 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.

## 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:

### Conjunction with other properties

Important conjunctions of normality with other subgroup properties are in the table below:[SHOW MORE]

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]

In some cases, we are interested in studying normal subgroups with the big group constrained to satisfy some group property. For instance:

### 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.
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

## Facts

### Isomorphism theorems

Theorem name/number Statement
First isomorphism theorem If $\varphi:G \to H$ is a surjective homomorphism, then the kernel $N$ of $\varphi$ 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 $N,H \le G$ such that $N$ is contained in the normalizer of $H$, then $N$ is normal in $NH$ and $NH/N \cong H/(H \cap N)$.
Third isomorphism theorem If $H \le K \le G$ 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 $G$, 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 Given $H \le G$, if $G$ is abelian, then $H$ is normal in $G$.
Transitive subgroup property No Normality is not transitive Can have $H \le K \le G$, $H$ normal in $K$, $K$ is normal in $G$, $H$ 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 Given $H_i, i \in I$, all normal in $G$, so is $\bigcap_{i \in I} H_i$
Strongly join-closed subgroup property Yes Normality is strongly join-closed Given $H_i, i \in I$, all normal in $G$, so is $\langle H_i \rangle_{i \in I}$
Quotient-transitive subgroup property Yes Normality is quotient-transitive $H \le K \le G$, $H$ normal in $G$, $K/H$ normal in $G/H$, then $K$ normal in $G$
Intermediate subgroup condition Yes Normality satisfies intermediate subgroup condition $H \le K \le G$, $H$ normal in $G$, then $H$ normal in $K$
Transfer condition Yes Normality satisfies transfer condition $H, K \le G$, $H$ normal in $G$, then $H \cap K$ normal in $K$
Image condition Yes Normality satisfies image condition $H$ normal in $G$, $\varphi:G \to K$ surjective, then $\varphi(H)$ normal in $K$
Inverse image condition Yes Normality satisfies inverse image condition $H$ normal in $G$, $\varphi:K \to G$ homomorphism, then $\varphi^{-1}(H)$ is normal in $K$
Upper join-closed subgroup property Yes Normality is upper join-closed $H \le G$, $K_i, i \in I$ intermediate subgroups, $H$ normal in each, then $H$ is normal in their join
Commutator-closed subgroup property Yes Normality is commutator-closed $H, K \le G$ both normal, then $[H,K]$ normal in $G$
Centralizer-closed subgroup property Yes Normality is centralizer-closed $H$ normal in $G$, then $C_G(H)$ also normal in $G$
Direct product-closed subgroup property Yes Normality is direct product-closed $H_i$ normal in $G_i$ implies direct product of $H_i$s normal in direct product of $G_i$s.
Arguesian subgroup property Yes NOrmality is Arguesian The collection of normal subgroups of a group form an Arguesian lattice.

## 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

## Testing

### The testing problem

Further information: Normality testing problem

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

### GAP command

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

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);

## References

### Textbook references

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