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From Groupprops

Names in other languages:German: Charakteristische Untergruppe; French: Sous-groupe caractéristique; Italian: Sottogruppo caratteristico

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Contents 1 History 2 Definition 2.1 Symbol-free definition 2.2 Definition with symbols 3 Importance 4 Formalisms 4.1 Second-order description 4.2 Function restriction expression 4.3 Relation implication formalism 4.4 Importance 5 Relation with other properties 5.1 Analogues 5.2 Stronger properties 5.3 Weaker properties 5.4 Relation with normality 6 Metaproperties 6.1 Transitivity 6.2 Trimness 6.3 Intersection-closedness 6.4 Join-closedness 6.5 Quotient-transitivity 6.6 Intermediate subgroup condition 7 Effect of property operators 7.1 The potentially operator 7.2 The intermediately operator 7.3 The simple group operator 8 Testing 8.1 The testing problem 8.2 GAP command for this subgroup property 9 Study of the notion 9.1 Mathematical subject classification 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

This is a variation of normality
View a complete list of variations of normality OR read a survey article on varying normality

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

History

This term was introduced by: Frobenius
View a complete list of terminology introduced by Frobenius

The notion of characteristic subgroup was introduced by Frobenius in 1895. His motivation was to capture the property of being a subgroup that is invariant under all symmetries of the group, and is hence intrinsic to the group. Frobenius wanted to use the term invariant subgroup but at the time, the term invariant subgroup was used for normal subgroup.

Definition

QUICK PHRASES: invariant under all automorphisms, automorphism-invariant, strongly normal, normal under outer automorphisms

Symbol-free definition

A subgroup of a group is termed characteristic or automorphism-invariant if it satisfies the following equivalent conditions:

• Every automorphism of the whole group takes the subgroup to within itself
• Every automorphism of the group restricts to an endomorphism of the subgroup
• Every automorphism of the group restricts to an automorphism of the subgroup

Definition with symbols

A subgroup H of a group G is termed characteristic or automorphism-invariant (in symbols, ^Notations) if it satisfies the following equivalent conditions:

• For any automorphism φ of G, . More explicitly, for any and ,
• For every automorphism φ of G, φ(H) = H.

Importance

Characteristic subgroups are important because they are genuinely invariant, not just under inner automorphisms, but under all automorphisms. In particular, every subgroup-defining function gives rise to a characteristic subgroup.

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)

Second-order description

This subgroup property is a second-order subgroup property, viz it has a second-order description in the theory of groups

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

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

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

Automorphism → Function

In other words, every automorphism of the whole group restricts to a well-defined function from the subgroup to itself.

• As the endo-invariance property with respect to the property of being an automorphism, viz:

Automorphism Endomorphism

In other words, every automorphism of the group restricts to an endomorphism of the subgroup.

• As the balanced subgroup property (function restriction formalism) with respect to automorphisms, viz:

Automorphism → Automorphism

In other words, every automorphism of the whole group restricts to an automorphism of the subgroup.

Relation implication formalism

Characteristicity can be expressed in the relation implication formalism with the left side being automorphs viz subgroups resembling each other via an automorphism of the whole group) and the right side being equal subgroups. In other words, a subgroup is characteristic if and only if every subgroup equivalent to it in the sense of being an automorph, is actually equal to it.

Importance

Characteristicity is important because all subgroup-defining functions give rise to characteristic subgroups.

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:

• Category: Variations of characteristicity
• Category: Opposites of characteristicity
• Category: Normal-to-characteristic subgroup properties

Analogues

A list of analogues of the property of being a characteristic subgroup, in other structures, is at:

Category:Analogues of characteristic subgroup

Stronger properties

• Fully characteristic subgroup
• Strictly characteristic subgroup
• Elementarily characteristic subgroup
• Intermediately characteristic subgroup
• Verbal subgroup

Weaker properties

• Normal 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
• Subnormal subgroup
• Potentially characteristic subgroup

Relation with normality

Check out: Characteristic versus normal

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 transitive: a subgroup with this property in a subgroup with this property, also has this property
View a complete list of transitive subgroup properties

The property of characteristicity is transitive. In other words, if H is a characteristic subgroup of K and K is a characteristic subgroup of G, then H is characteristic in G. This follows from the fact that characteristicity is a balanced subgroup property with respect to the function restriction formalism, and any balanced subgroup property is transitive.

For full proof, refer: Characteristicity is 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

The trivial subgroup is characteristic, and so is the whole group (for obvious reasons).

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

Since characteristicity is an invariance property with respect to the function restriction formalism, it is intersection-closed. That is, the intersection of a family of characteristic subgroups is characteristic.

For full proof, refer: Characteristicity is strongly 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 characteristicity is an invariance property with respect to a property stronger than that of being an endomorphism, the property of being characteristic is join-closed. That is, an arbitrary join of characteristic subgroups is characteristic.

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

Quotient-transitivity

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

The subgroup property of being characteristic is a quotient-transitive subgroup property. That is, if are groups such that H is characteristic in G and K / H is characteristic in G / H, then K is characteristic in G.

Template:Not dirprodclosedsgp

If H₁ is characteristic in G₁ and H₂ is characteristic in G₂, it is not necessary that be characteristic in G₁ × G₂. A trivial counterexample is to set G₁ = G₂ = G and H₁ = G and H₂ trivial. Here, the direct product is G₁ itself, but we know that the coordinate exchange automorphism takes G₁ to G₂, hence G₁ cannot be characteristic.

Intermediate subgroup condition

This subgroup property does not satisfy the intermediate subgroup condition

If H is a characteristic subgroup of G, and K is an intermediate subgroup (i.e. ) then H need not be characteristic in K.

Effect of property operators

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)

The potentially operator

Applying the potentially operator to this property gives: potentially characteristic subgroup

Applying the potentially operator to the subgroup property of being characteristic gives the subgroup property of being potentially characteristic. A subgroup H is said to be potentially characteristic in a group G if there exists a group K containing G such that H is characteristic in K.

The intermediately operator

Applying the intermediately operator to this property gives: intermediately characteristic subgroup

Applying the intermediately operator to the subgroup property of being characteristic gives the subgroup property of being intermediately characteristic. A subgroup H of a group G is termed intermediately characteristic in G if for every intermediate subgroup K of G containing H, H is characteristic in K.

The simple group operator

Applying the simple group operator to this property gives: characteristically simple group

Applying the simple group operator to the subgroup property of characteristicity gives the group property of being characteristically simple.

Testing

The testing problem

Further information: characteristicity testing problem

Given generating sets for a group and a subgroup, the problem of determining whether the subgroup is characteristic in the group cannot be solved directly. However, it can be reduced to the problem of finding a small generating set for the automorphism group of the bigger group.

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:IsCharacteristicSubgroup
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 characteristic in a group is:

IsCharacteristicSubgroup (group, subgroup);

where

subgroup

and

group

may be defined on the spot in terms of generators or may refer to things defined previously.

Study of the notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20A05

References

Textbook references

• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell^{More info}, Page 17 (formal definition, in paragraph)
• Abstract Algebra by David S. Dummit and Richard M. Foote^{More info}, Page 135 (formal definition)
• Topics in Algebra by I. N. Herstein^{More info}, Page 70, Problem 7(a)

External links

Search for "characteristic+subgroup" on the World Wide Web:
Scholarly articles: Google Scholar, JSTOR
Books: Google Books, Amazon
This wiki: Internal search, Google site search
Encyclopaedias: Wikipedia (or using Google), Citizendium
Math resource pages:Mathworld, Planetmath, Springer Online Reference Works
Math wikis: Topospaces, Diffgeom, Commalg, Noncommutative
Discussion fora: Mathlinks, Google Groups
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Definition links

• Wikipedia page
• Planetmath page
• Springer Online Reference page