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Normal subgroup

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

Use Google translate to translate this page to French, German, Spanish, Italian

1 History
- 1.1 Origin of the concept
- 1.2 Origin of the term
2 Definition
3 Importance
4 Examples
- 4.1 Examples
- 4.2 Non-examples
5 Formalisms
6 Relation with other properties
7 Metaproperties
8 Effect of property operators
9 Testing
- 9.1 The testing problem
- 9.2 GAP command for this subgroup property
10 References
- 10.1 Textbook references
11 External links
- 11.1 Definition links

This article is about a basic definition in group theory. The article text may, however, contain more material. Rate its utility as a basic definition article on the talk page
View a complete list of basic definitions in group theory OR Go through a guided tour for beginners to this wiki

This article defines a subgroup property that is pivotal (viz important) among existing subgroup properties
View a list of pivotal subgroup properties OR View a complete list of subgroup properties

For survey articles related to this, refer: Category:Survey articles related to normality

This subgroup property is always true for a subgroup of an Abelian group
View other such properties

History

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
View a complete list of terminology 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.

Definition

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

Symbol-free definition

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

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).
It is the kernel of a homomorphism from the group.
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).
Its left cosets are the same as its right cosets (that is, it commutes with every element of the group)

Definition with symbols

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

For all $g$ in $G$ , $gNg^{-1} \subseteq N$ . More explicitly, for all $g \in G, h \in N$ , we have $ghg^{-1} \in N$ .
There is a homomorphism $φ$ from $G$ to another group such that the kernel of $φ$ is precisely $N$ .
For all $g$ in $G$ , $g N g - 1 = N$ .
For all $g$ in $G$ , $g N = N g$ .

Equivalence of definitions

The equivalence of all definitions except the second one is more or less direct. The equivalence between the first two definitions is available at: normal subgroup equals kernel of homomorphism.

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

If you're interested in normal subgroups in a particular group, view the article on that particular group and hunt for the subsection titled Normal subgroups

Examples

In an Abelian group, every subgroup is normal (there are non-Abelian groups, such as the quaternion group, where every subgroup is normal).
If $G$ is an internal direct product of subgroups $H$ and $K$ , both $H$ and $K$ are normal in $G$ .
The center, commutator subgroup are normal.

Non-examples

Here are some examples of non-normal subgroups:

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).
More generally, in any dihedral group, the two-element subgroup generated by a reflection is not normal.
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:

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

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

Function restriction expression

This subgroup property can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
View other properties expressible in this formalism OR View the function restriction formalism chart for a graphic placement of this property

Normality can be expressed in terms of the function restriction formalism in any of the following ways:

As the invariance property with respect to the function property of being an inner automorphism, viz:

Inner automorphism $\to$ Function

In other words, a subgroup is normal if and only if every inner automorphism of the whole group, when restricted to the subgroup, defines a function from the subgroup to itself

Via the function restriction expression with the left side being inner automorphisms and the right side being endomorphisms, viz:

Inner automorphism $\to$ Endomorphism

In other words, a subgroup is normal if and only if every inner automorphism of the whole group, when restricted to the subgroup, defines an endomorphism on the subgroup.

Via the function restriction expression with the left side being inner automorphisms and the right side being automorphisms, viz:

Inner automorphism $\to$ Automorphism

In other words, a subgroup is normal if and only if every inner automorphism of the group, when restricted to the subgroup, defines an automorphism of the subgroup.

Relation implication expression

This subgroup property can be defined and viewed using a relation implication expression
View all subgroup properties having such expressions

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

In the general language of a variety of algebras with zero, the property of normality translates to the property of being an ideal.

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

Characteristic subgroup: For proof of the implication, refer Characteristic implies normal and for proof of its strictness (i.e. the reverse implication being false) refer Normal not implies characteristic
Direct factor:For proof of the implication, refer Direct factor implies normal and for proof of its strictness (i.e. the reverse implication being false) refer Normal not implies direct factor
Central factor: For proof of the implication, refer Central factor implies normal and for proof of its strictness (i.e. the reverse implication being false) refer Normal not implies central factor
Transitively normal subgroup
Central subgroup
Cocentral subgroup
Fully characteristic subgroup

Conjunction with other properties

Important conjunctions of normality with other subgroup properties:

Conjugacy-closed normal subgroup: Conjunction of being conjugacy-closed and normal.
Normal CEP-subgroup: Conjunction of being normal and a CEP-subgroup. This turns out to be the same as the property of being a transitively normal subgroup
Normal Hall subgroup: Conjunction of being normal and a Hall subgroup.

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.

Abelian normal subgroup: Conjunction of being an Abelian group and a normal subgroup.
Solvable normal subgroup: Conjunction of being a solvable group and a normal subgroup.
Nilpotent normal subgroup: Conjunction of being a nilpotent group and a normal subgroup.
Perfect normal subgroup: Conjunction of being a perfect group and a normal subgroup.
Simple normal subgroup: Conjunction of being a simple group and a normal subgroup.

Weaker properties

Subnormal subgroup: Obtained by applying the subordination operator to normality. For proof of the implication, refer Normal implies subnormal and for proof of its strictness (i.e. the reverse implication being false) refer Subnormal not implies normal
Pronormal subgroup: For proof of the implication, refer Normal implies pronormal and for proof of its strictness (i.e. the reverse implication being false) refer Pronormal not implies normal
Ascendant subgroup
Descendant subgroup
Serial subgroup
Permutable subgroup: For proof of the implication, refer Normal implies permutable and for proof of its strictness (i.e. the reverse implication being false) refer Permutable not implies normal
S-quasinormal subgroup was introduced by Kegel. A subgroup H of G is called S-quasinormal if H permutes all Sylow subgroups of G.
SS-quasinormal subgroup was introduced by Zhencai Shen and Shirong Li, A subgroup H of G is called SS-quasinormal if there is a subgroup B such that G=HB and H permutes all Sylow subgroups of B.

Related operators

Lists are available at:

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)

Transitivity

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

Normality is not transitive. That is, it is possible to have groups $G$ ≤ $H$ ≤ $K$ such that $G$ is normal in $H$ and $H$ is normal in $K$ but $G$ is not normal in $K$ .

For full proof, refer: Normality is not transitive

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 all trim subgroup properties OR view trivially true subgroup properties OR view identity-true subgroup properties

Normality is a trim subgroup property: both the trivial subgroup and the improper subgroup are normal as subgroups of the whole group.

Intersection-closedness

This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property
View a complete list of intersection-closed subgroup properties

On account of normality being an invariance property, it is intersection-closed, viz an arbitrary intersection of normal subgroups is again normal in the whole group. For full proof, refer: Normality is intersection-closed

Join-closedness

This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property
View a complete list of join-closed subgroup properties

Since normality is an invariance property with respect to functions that are all endomorphisms, it is also join-closed, viz the subgroup generated by an arbitrary family of normal subgroups is again normal. For full proof, refer: Normality is join-closed

Intermediate subgroup condition

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
View all subgroup properties satisfying 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

Transfer condition

This subgroup property satisfies the transfer condition

Normality satisfies the transfer condition. In other words, if $H$ ≤ $G$ 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

Normality satisfies the inverse image condition. That is, if $φ$ is a homomorphism of groups, then the inverse image via $φ$ of any normal subgroup on the right is a normal subgroup on the left.

For full proof, refer: Normality satisfies inverse image condition

Quotient-transitivity

This subgroup property is quotient-transitive, viz the corresponding quotient property is transitive.

The property of normality is a quotient-transitive subgroup property. That is, if $H$ ≤ $K$ ≤ $G$ 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

Image condition

This subgroup property satisfies the image condition, viz under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property

Under any surjective homomorphism, the image of a normal subgroup is normal.

For full proof, refer: Normality satisfies image condition

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

Upper join-closedness

This subgroup property is upper join-closed, viz if a subgroup has the property in two intermediate subgroups, it also has the property in their join

If $H$ is a subgroup of $G$ , and $K 1, K 2$ are intermediate subgroups such that $H \triangleleft K_1, K_2$ , then $H \triangleleft <K_1,K_2>$ .

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

Effect of property operators

The left transiter

Applying the left transiter to this property gives: characteristic subgroup

The left transiter of normality is the property of being characteristic. 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

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 quotientable automorphisms.

Subordination and related operators

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 but is unlikely to be an invariance property.
The ascendant closure is the property of being an ascendant subgroup
The descendant closure is the property of being a descendant subgroup
The serial closure is the property of being a serial subgroup

Hereditarily operator

The hereditarily operator

Applying the hereditarily operator to this property gives: hereditarily normal subgroup

The result of applying the left-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 $G$ .

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 operator

Applying the maximal operator to this property gives: maximal 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 for this subgroup property

This subgroup property can be tested using Groups, Algorithms, Programming (GAP). The GAP command for testing this subgroup property is:IsNormal
View a complete list of subgroup properties testable with commands in GAP OR View subgroup properties codable in GAP or 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^{More info}, Page 6 (first use, single definition as part of a paragraph)
Abstract Algebra by David S. Dummit and Richard M. Foote^{More info}, Page 80 (first use), Page 82 (formal definition, and Theorem 6 giving equivalent formulations)
Topics in Algebra by I. N. Herstein^{More info}, Page 50 (formal definition, as part of Section 2.6)
A Course in the Theory of Groups by Derek J. S. Robinson^{More info}, Page 15 (definition by proposition 1.3.15)
Algebra by Serge Lang^{More info}, Page 14 (definition in paragraph)

External links

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Definition links

Retrieved from "http://groupprops.wiki-site.com/index.php/Normal_subgroup"

Normal subgroup

From Groupprops

Contents

History

Origin of the concept

Origin of the term

Definition

Symbol-free definition

Definition with symbols

Equivalence of definitions

Importance

Examples

Examples

Non-examples

Formalisms

First-order description

Function restriction expression

Relation implication expression

Variety formalism

Relation with other properties

Stronger properties

Conjunction with other properties

Weaker properties

Related operators

Metaproperties

Transitivity

Trimness

Intersection-closedness

Join-closedness

Intermediate subgroup condition

Transfer condition

Inverse image condition

Quotient-transitivity

Image condition

Direct product-closedness

Arguesianness

Upper join-closedness

Effect of property operators

The left transiter

The right transiter

Subordination and related operators

The subordination operator

Hereditarily operator

The hereditarily operator

The upward-closure operator

The maximal operator

Testing

The testing problem

GAP command for this subgroup property

References

Textbook references

External links

Definition links

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