Normal subgroup

From Groupprops

(Redirected from Monomial automorphism-invariant subgroup)

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

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1 History
- 1.1 Origin of the concept
- 1.2 Origin of the term
2 Definition
3 Importance
4 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
10 References
- 10.1 Textbook references
11 External links
- 11.1 Definition links

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This article defines a subgroup property that is pivotal (viz important) among existing subgroup properties
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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

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, 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:

It is the kernel of a homomorphism from the group.
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 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).
It is a union of conjugacy classes.
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, $N \triangleleft G$ or $G \triangleright N$ ^Notations) if the following equivalent conditions hold:

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

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

Extreme examples

The trivial subgroup is always normal.
Every group is normal as a subgroup of itself.

Examples

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).
If $G$ is an internal direct product of subgroups $H$ and $K$ , both $H$ and $K$ are normal in $G$ .
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.

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.
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:

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

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 has a number of function restriction expressions:

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. This function restriction expression for normality is right tight -- we cannot replace automorphisms by any stronger notion. Further information: Inner automorphism to automorphism is right tight for normality

Via the function restriction expression with the left side being class-preserving automorphisms and the right side being automorphisms:

Class-preserving automorphism $\to$ Automorphism

In other words, a subgroup is normal if and only if every class-preserving automorphism of the group (i.e., every automorphism preserving conjugacy classes) restricts to an automorphism of the subgroup.

Via the function restriction expression with the left side being subgroup-conjugating automorphisms and the right side being automorphisms:

Subgroup-conjugating automorphism $\to$ Automorphism

Via the function restriction expression with the left side being normal automorphisms and the right side being automorphisms:

Normal automorphism $\to$ Automorphism

This function restriction expression is both left and right tight for normality. Further information: Normal automorphism to automorphism is left tight for normality

For more function restriction expressions for normality, check out Funky definitions of normal subgroup#Definition via function restriction expression.

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

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

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.; View a survey article comparing and contrasting these terms, at: Characteristic versus normal
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.; View a survey article comparing and contrasting these terms, at: Direct factor versus normal
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.; View a survey article comparing and contrasting these terms, at: Central factor versus normal
Transitively normal subgroup
Central subgroup
Cocentral subgroup
Fully characteristic subgroup

View a more comprehensive list of subgroup properties stronger than normality

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.

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.

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.

View a complete list of conjunctions of normality with group properties

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.: View a survey article comparing and contrasting these terms, at: normal versus subnormal
2-subnormal subgroup
3-subnormal subgroup
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.; View a survey article comparing and contrasting these terms, at: normal versus permutable
Conjugate-permutable subgroup

View a more comprehensive list of subgroup properties weaker than normality

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.

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)

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
View a complete list of transitive subgroup properties|View facts related to transitivity of subgroup properties

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 the intermediate subgroup condition|View facts related to the 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: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
View a complete list of such properties

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: 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 $H$ ≤ $K$ ≤ $G$ 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, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property
View a complete list of subgroup properties satisfying the image condition

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

For full proof, refer: Normality satisfies 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 a complete list of 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

Commutator-closedness

This subgroup property is commutator-closed: the commutator of two subgroups each with the property, also has the property
View a complete list of commutator-closed subgroup properties

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

For full proof, refer: Normality is commutator-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

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
View other such properties

If $H$ is a subgroup of $G$ , and $K_i, i \in I$ are intermediate subgroups such that $H \triangleleft K_i$ for all $i \in I$ , then $H \triangleleft \langle K_i \rangle_{i \in I}$ .

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 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 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

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 proper operator

Applying the maximal proper operator to this property gives: maximal normal subgroup

The minimal operator

Applying the minimal operator to this property gives: minimal 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 commands|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, ISBN 0471433349, 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

Facts about Normal subgroupRDF feed

Applying operator gives	Characteristic subgroup +, Transitively normal subgroup +, Subnormal subgroup +, Hereditarily normal subgroup +, Upward-closed normal subgroup +, Maximal normal subgroup +, and Minimal normal subgroup +
Defined in	AlperinBell (6, ?, single definition as part of a paragraph) +, DummitFoote (82, ?, formal definition, and Theorem 6 giving equivalent formulations) +, Herstein (50, Section 2.6, formal definition) +, RobinsonGT (15, Proposition 1.3.15, definition introduced through proposition) +, Lang (14, ?, definition in paragraph) +, Artin (52, Point (4.8), formal definition, followed by equivalent definition-cum-proposition in (4.9)) +, Wikipedia (?, ?, ?) +, Planetmath (?, ?, ?) +, Mathworld (?, ?, ?) +, Springer Online Reference Works (?, ?, ?) +, and Citizendium (?, ?, ?) +
Defining ingredient	Kernel +, Homomorphism +, Inner automorphism +, Conjugate subgroups +, Left coset +, Right coset +, and Conjugacy class +
Referenced in	AlperinBell (6, ?, single definition as part of a paragraph) +, DummitFoote (82, ?, formal definition, and Theorem 6 giving equivalent formulations) +, Herstein (50, Section 2.6, formal definition) +, RobinsonGT (15, Proposition 1.3.15, definition introduced through proposition) +, Lang (14, ?, definition in paragraph) +, Artin (52, Point (4.8), formal definition, followed by equivalent definition-cum-proposition in (4.9)) +, Wikipedia (?, ?, ?) +, Planetmath (?, ?, ?) +, Mathworld (?, ?, ?) +, Springer Online Reference Works (?, ?, ?) +, and Citizendium (?, ?, ?) +
Stronger than	Subnormal subgroup +, 2-subnormal subgroup +, 3-subnormal subgroup +, Pronormal subgroup +, Ascendant subgroup +, Descendant subgroup +, Serial subgroup +, Permutable subgroup +, and Conjugate-permutable subgroup +
Term introduced by	Galois +
Weaker than	Characteristic subgroup +, Direct factor +, Central factor +, Transitively normal subgroup +, Central subgroup +, Cocentral subgroup +, Fully characteristic subgroup +, Conjugacy-closed normal subgroup +, Normal Hall subgroup +, Abelian normal subgroup +, Solvable normal subgroup +, Nilpotent normal subgroup +, Perfect normal subgroup +, and Simple 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

Extreme 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

Centralizer-closedness

Commutator-closedness

Direct product-closedness

Arguesianness

Upper join-closedness

Effect of property operators

The left transiter

The right transiter

The subordination operator

The hereditarily operator

The upward-closure operator

The maximal proper operator

The minimal operator

Testing

The testing problem

GAP command

References

Textbook references

External links

Definition links

Views

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