Normal subgroup

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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This article defines a subgroup property that is pivotal (viz important) among existing subgroup properties
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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

Symbol-free definition

A subgroup of a group is said to be normal if it satisfies the following equivalent conditions:

1. (Homomorphism kernel definition): It is the kernel of a homomorphism from the group.
2. (Inner automorphisms definition): It is invariant under all inner automorphisms. 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 definition): It equals each of its conjugates in the whole group. This definition also motivates the term self-conjugate subgroup for normal subgroup (which was used earlier).
4. (Cosets definition): Its left cosets are the same as its right cosets (that is, it commutes with every element of the group).
5. (Conjugacy class definition): It is a union of conjugacy classes.
6. (Commutator definition): It contains its commutator with the whole group.

Definition with symbols

A subgroup $N$ of a group $G$ is said to be normal in $G$ (in symbols, $N◃G$ or $G▹N$^Notations) if the following equivalent conditions hold:

1. (Homomorphism kernel definition): There is a homomorphism $\phi$ from $G$ to a group $K$ such that the kernel of $\phi$ is precisely $N$. In other words, $\phi (x)$ is the identity element of $K$ if and only if $x\in N$.
2. (Inner automorphisms definition): For all $g\in G$, $gN{g}^{-1}\subseteq N$. More explicitly, for all $g\in G,h\in N$, we have $gh{g}^{-1}\in N$.
3. (Equals conjuate definition): For all $g$ in $G$, $gN{g}^{-1}=N$.
4. (Cosets definition): For all $g$ in $G$, $gN=Ng$.
5. (Conjugacy classes definition): $N$ is a union of conjugacy classes.
6. (Commutator definition): The commutator $[N,G]$ (which coincides with the commutator $[G,N]$) is contained in $N$.

Equivalence of definitions

• For the equivalence between definitions (1) and (2), refer Normal subgroup equals kernel of homomorphism.
• The equivalence between definitions (2) and (3) follows from a more general fact: 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
• The equivalence between definitions (3) and (4) is a direct manipulation of equations involving elements and subsets. Further information: manipulating equations in groups
• The equivalence of definitions (2) or (3) with definition (5) is a straightforward unraveling of the meaning of conjugacy class.
• The equivalence of definitions (2) or (3) with definition (6) involves a straightforward unraveling of the meaning of commutator, along with a little bit of manipulation. 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: <random>The definition of normality in terms of the commutator (definition (6)) doesn't usually make it to most elementary textbook treatments of normality. But it is a very important way of thinking about normal subgroups. Here's a somewhat stronger cool fact: If

$[G,H]\le K\le H\le G$

, then

$K$

is normal in

$G$

.@@@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.@@@There's something special about groups that makes it true that the kernel of a homomorphism is actually a subgroup. Learn more at ideals are subalgebras in the variety of groups and characteristic subalgebras are ideals in the variety of groups</random>

History

[SHOW MORE]

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.

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.

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: $N$ is normal in $G$ if and only if:

$\mathrm{\forall }g\in G,h\in N:gh{g}^{-1}\in N$

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 $⟹$ 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:

• 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	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions	comparison
characteristic subgroup	invariant under all automorphisms	characteristic implies normal	normal not implies characteristic (see also list of examples)	|FULL LIST, MORE INFO	characteristic versus normal
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)	|FULL LIST, MORE INFO	central factor versus normal
direct factor	factor in internal direct product	direct factor implies normal	normal not implies direct factor (see also list of examples)	|FULL LIST, MORE INFO	direct factor versus normal
central subgroup	contained in the center	central implies normal	normal not implies central (see also list of examples)	Abelian normal subgroup|FULL LIST, MORE INFO	normal subgroup versus central subgroup

[SHOW MORE]

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
fully invariant subgroup	invariant under all endomorphisms	(via characteristic)	(via characteristic)(see also list of examples)	Characteristic subgroup|FULL LIST, MORE INFO	--
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, Isomorph-normal characteristic subgroup, Isomorph-normal subgroup|FULL LIST, MORE INFO
isomorph-containing subgroup	contains all isomorphic subgroups	(via characteristic)	(via characteristic) (see also list of examples)	Characteristic 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
cocentral subgroup	product with center is whole group	cocentral implies normal	normal not implies cocentral	|FULL LIST, MORE INFO	--
SCAB-subgroup	subgroup-conjugating automorphism restricts to subgroup-conjugating automorphism of subgroup			|FULL LIST, MORE INFO	--
conjugacy-closed normal subgroup	conjugacy-closed subgroup and normal subgroup			|FULL LIST, MORE INFO	--
abelian-quotient subgroup	contains the commutator 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:[SHOW MORE]

Conjunction	Other component of conjunction	Intermediate notions	Additional comments
conjugacy-closed normal subgroup	conjugacy-closed subgroup	|FULL LIST, MORE INFO
transitively normal subgroup	CEP-subgroup	|FULL LIST, MORE INFO
normal Sylow subgroup	Sylow subgroup	Fully invariant subgroup|FULL LIST, MORE INFO
normal Hall subgroup	Hall subgroup	Fully invariant 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:[SHOW MORE]

Conjunction	Other component of conjunction	Intermediate notions	Additional comments
abelian normal subgroup	abelian group	|FULL LIST, MORE INFO
* cyclic normal subgroup	cyclic group	Abelian normal subgroup|FULL LIST, MORE INFO
* solvable normal subgroup	solvable group	|FULL LIST, MORE INFO
* nilpotent normal subgroup	nilpotent group	|FULL LIST, MORE INFO
simple normal subgroup	simple group	|FULL LIST, MORE INFO
* perfect normal subgroup	perfect group	|FULL LIST, MORE INFO
finite normal subgroup	finite group	|FULL LIST, MORE INFO
* periodic normal subgroup	periodic group	|FULL LIST, MORE INFO
* finitely generated normal subgroup	finitely generated group	|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:[SHOW MORE]

Conjunction	Property of the group	Intermediate notions	Additional comments
normal subgroup of finite group	finite group	|FULL LIST, MORE INFO
* normal subgroup of finitely generated group	finitely generated group	|FULL LIST, MORE INFO
* normal subgroup of periodic group	periodic group	|FULL LIST, MORE INFO
normal subgroup of group of prime power order	group of prime power order	|FULL LIST, MORE INFO

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions	comparison
Subnormal subgroup	obtained using subordination operator on normality	normal implies subnormal	subnormal not implies normal (see also list of examples)	2-subnormal subgroup|FULL LIST, MORE INFO	contrasting subnormality of various depths
Permutable subgroup (also called quasinormal subgroup)	permutes with every subgroup	normal implies permutable	permutable not implies normal (see also list of examples)	|FULL LIST, MORE INFO
2-subnormal subgroup	normal subgroup of normal subgroup		normality is not transitive (see also list of examples)	|FULL LIST, MORE INFO	(above)

[SHOW MORE]

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions	comparison
3-subnormal subgroup	normal of normal of normal			2-subnormal subgroup|FULL LIST, MORE INFO	contrasting subnormality of various depths
4-subnormal subgroup	2-subnormal of 2-subnormal			2-subnormal subgroup|FULL LIST, MORE INFO	contrasting subnormality of various depths
Ascendant subgroup				Subnormal subgroup|FULL LIST, MORE INFO
Descendant subgroup				Subnormal subgroup|FULL LIST, MORE INFO
Serial subgroup				Subnormal subgroup|FULL LIST, MORE INFO
Hypernormalized subgroup	iterated normalizers reach whole group		(see also list of examples)	|FULL LIST, MORE INFO
Finitarily hypernormalized subgroup	iterated normalizers reach whole group in finite number of steps			|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)	|FULL LIST, MORE INFO	subnormal-to-normal and normal-to-characteristic
Weakly pronormal subgroup		(via pronormal)		|FULL LIST, MORE INFO	(above)
Paranormal subgroup		(via pronormal)		|FULL LIST, MORE INFO	(above)
Weakly normal subgroup		(via pronormal)		|FULL LIST, MORE INFO	(above)
Intermediately subnormal-to-normal subgroup		(via pronormal)	(see also list of examples)	|FULL LIST, MORE INFO	(above)
Conjugate-permutable subgroup	permutes with its conjugate subgroups			2-subnormal subgroup|FULL LIST, MORE INFO
Modular subgroup				|FULL LIST, MORE INFO
Automorph-permutable subgroup				|FULL LIST, MORE INFO
Elliptic 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:

• Normal core: This takes a subgroup and outputs the largest normal subgroup inside it.
• Normal closure: This takes a subgroup and outputs the smallest normal subgroup containing it.
• Normalizer: This takes a subgroup and outputs the largest subgroup within which it is normal.

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

• Normal core of normalizer
• Hypernormalizer

Further information: Category: Subgroup operators related to normality

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 $H\le K\le G$, $H$ 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 $⟨H_i{⟩}_{iinI}$
Quotient-transitive subgroup property	Yes	Normality is quotient-transitive	$H\le K\le G$, $H$ normal in $G$, $K/H$ normal in $G$, 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 $G$, $\phi :G\to K$ surjective, then $\phi (H)$ normal in $K$
Inverse image condition	Yes	Normality satisfies inverse image condition	$H$ normal in $G$, $\phi :K\to G$ homomorphism, then ${\phi }^{-1}(H)$ 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.

For more information on these metaproperties: [SHOW MORE]

Tautology when whole group is abelian

This subgroup property is an abelian-tautological subgroup property: it is always true for a subgroup of an abelian group.
View a complete list of abelian-tautological 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 non-abelian 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 transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View 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 $H\le K\le G$ such that $H$ is normal in $K$ and $K$ is normal in $G$ but $H$ is not normal in $G$. The smallest counterexample is where $G$ is the dihedral group of order eight, $H$ is a non-normal subgroup of order two and $K$ is a Klein four-subgroup 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 identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true 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 identity-true, Endo-invariance implies trim

Intersection-closedness

YES: This subgroup property is intersection-closed: 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 intersection-closed subgroup property.
ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closed
ABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness

On account of normality being an invariance property, it is strongly intersection-closed, 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 $H_i,i\in I$ is a collection of normal subgroups of $G$, then ${\bigcap }_{i\in I}{H}_i$ is normal in $G$.

For full proof, refer: Normality is strongly intersection-closed

Further information: Invariance implies strongly intersection-closed, Characteristicity is strongly intersection-closed, Full invariance is strongly intersection-closed

Join-closedness

YES: This subgroup property is join-closed: 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 join-closed subgroup property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

Since normality is an invariance property with respect to functions that are all endomorphisms, it is also strongly join-closed, 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 $H_i,i\in I$ is a collection of normal subgroups of $G$, then the join $⟨H_i{⟩}_{i\in I}$ is normal in $G$.

For full proof, refer: Normality is strongly join-closed

Further information: Endo-invariance implies strongly join-closed, Characteristicity is strongly join-closed, Full invariance is strongly join-closed

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 $H$ is a normal subgroup of $G$ and $K$ is a subgroup of $G$ containing $H$, then $H$ is normal in $K$. 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 left-inner subgroup property.

For full proof, refer: Normality satisfies intermediate subgroup condition

Further information: Left-inner implies intermediate subgroup condition, Left-extensibility-stable 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 $H\le G$ is normal, and $K$ is any subgroup of $G$ then $H\cap K$ is a normal subgroup of $K$.

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 $\phi :K\to G$ is a homomorphism and $H$ is a normal subgroup of $G$, then ${\phi }^{-1}(H)$ is a normal subgroup of $K$.

For full proof, refer: Normality satisfies inverse image condition

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties

The property of normality is a quotient-transitive subgroup property. That is, 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 $G$.

For full proof, refer: Normality is quotient-transitive

Further information: Quotient-balanced implies quotient-transitive, Characteristicity is quotient-transitive, Full invariance is quotient-transitive

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 $H$ is a normal subgroup of $G$ and $\phi :G\to K$ is a surjective homomorphism, then $\phi (H)$ is normal in $K$.

For full proof, refer: Normality satisfies image condition

Further information: Subnormality satisfies image condition, Pronormality satisfies image condition, Characteristicity does not satisfy image condition

Centralizer-closedness

This subgroup property is centralizer-closed: the centralizer of any subgroup with this property, in the whole group, again has this property
View other centralizer-closed subgroup properties

The centralizer of a normal subgroup is always normal. Thus, if $H$ is normal in $G$, so is $C_G(H)$. This follows from the general fact that normality can be described as an auto-invariance property: an invariance property with respect to a property of automorphisms (namely, inner automorphisms).

For full proof, refer: Normality is centralizer-closed

Further information: Auto-invariance implies centralizer-closed, Characteristicity is centralizer-closed, Automorph-conjugacy is centralizer-closed

Commutator-closedness

This subgroup property is commutator-closed: the commutator of two subgroups each with the property, also has the property.
View other commutator-closed subgroup properties

A commutator of two normal subgroups is normal. Thus, if $H,K$ are normal subgroups of $G$, so is $[H,K]$. 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 commutator-closed

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, Endo-invariance implies commutator-closed, Characteristicity is commutator-closed, Full invariance is commutator-closed

Upper join-closedness

This subgroup property is upper join-closed, 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 $H$ is a subgroup of $G$, and $K_i,i\in I$ are intermediate subgroups such that $H◃K_i$ for all $i\in I$, then $H◃⟨K_i{⟩}_{i\in I}$.

In fact, there is a unique largest subgroup of $G$ inside which $H$ is normal. This subgroup is termed the normalizer of $H$ in $G$.

For full proof, refer: Normality is upper join-closed

Further information: Characteristicity is not upper join-closed, 2-subnormality is not upper join-closed

Direct product-closedness

This subgroup property is direct product-closed: it is closed under taking arbitrary direct products of groups

If $I$ is a nonempty indexing set and $H_i$ is normal in $G_i$ for each $i\in I$, then the direct product of the $H_i$s is normal in the direct product of the $G_i$s.

For full proof, refer: Normality is direct product-closed

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, join-closed and intersection-closed).

For full proof, refer: Normality is Arguesian

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
subordination operator	normal subgroup of normal subgroup of ... of normal subgroup	subnormal subgroup
hereditarily operator	every subgroup of it is normal	hereditarily normal subgroup
upward-closure operator	every subgroup containing it is normal	upward-closed normal subgroup
maximal proper operator	proper normal subgroup contained in no other proper normal subgroup	maximal normal subgroup
minimal operator	nontrivial normal subgroup containing no other nontrivial normal subgroup	minimal normal subgroup

For more information on these property operators: [SHOW MORE]

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 $H$ is a subgroup of $G$ such that whenever $G$ is normal in some bigger group $K$ so is $H$, then $H$ is characteristic in $G$. Conversely, if $H$ is characteristic in $G$ and $G$ is normal in $K$, $H$ 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 upper-hook 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 subordination of normality is subnormality. This is also transitive and identity-true. A subnormal subgroup is thus a subgroup for which there exists an ascending chain of finite length from the subgroup to the whole group with each being normal in its successor. The subnormal depth of a subnormal subgroup is the length of the shortest possible chain for it. We say a subnormal subgroup has subnormal depth $d$ if its subnormal depth is at most $d$. Further information: Normality is not transitive, There exist subgroups of arbitrarily large subnormal depth
• The ascendant closure is the property of being an ascendant subgroup. An ascendant subgroup is a subgroup for which there exist an ascending, possibly transfinite, chain of subgroups with each being normal in its successor. There do exist ascendant subgroups that are not subnormal. Further information: Ascendant not implies subnormal
• The descendant closure is the property of being a descendant subgroup. A descendant subgroup is a subgroup for which there exists a descending, possibly transfinite, chain of subgroups with each being normal in its predecessor. There do exist descendant subgroups that are not subnormal. Further information: Descendant not implies subnormal
• The serial closure is the property of being a serial 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 $H$ of a group $G$ is termed hereditarily normal if every subgroup $N$ of $H$ is normal in $G$. 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 upward-closure operator

Applying the upward-closure operator to this property gives: upward-closed normal subgroup

The result of applying the upward closure operator to the subgroup property of being normal gives the property of being upward-closed normal, viz., a subgroup $H$ is upward-closed normal in $G$ if for any intermediate subgroup $K$ of $G$, $K$ is normal in $G$.

The commutator subgroup is an example of an upward-closed 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 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);

References

Textbook references

• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, Page 6, (single definition as part of a paragraph)^{More info}
• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, Page 82, (formal definition, and Theorem 6 giving equivalent formulations)^{More info}. Also, Page 80 (first use).
• Topics in Algebra by I. N. Herstein, Page 50, Section 2.6, (formal definition)^{More info}
• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, Page 15, Proposition 1.3.15, (definition introduced through proposition)^{More info}
• Algebra by Serge Lang, ISBN 038795385X, Page 14, (definition in paragraph)^{More info}
• Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, Page 52, Point (4.8), (formal definition, followed by equivalent definition-cum-proposition in (4.9))^{More info}

External links

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Definition links

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• Planetmath page
• Mathworld page
• Springer Online Reference page
• Citizendium page
• Differential geometry wiki page

Names in other languages:German: Normalteiler; French: Sous-groupe normal; Spanish: Subgrupo normal; Italian: Sottogruppo normale

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