Characteristic subgroup

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

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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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VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

This is a variation of normality|Find other variations of normality | 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: Ferdinand Georg 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:

1. Every automorphism of the whole group takes the subgroup to within itself
2. Every automorphism of the group restricts to an endomorphism of the subgroup
3. 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, $HcharG$^Notations) if it satisfies the following equivalent conditions:

1. For any automorphism $\phi$ of $G$, $\phi (H)\subseteq H$. More explicitly, for any $h\in H$ and $\phi \in Aut(G)$, $\phi (h)\in H$
2. For every automorphism $\phi$ of $G$, $\phi (H)\subseteq H$ and $\phi$ restricts to an endomorphism of $H$.
3. For every automorphism $\phi$ of $G$, $\phi (H)=H$ and $\phi$ restricts to an automorphism of $H$.

Equivalence of definitions

The equivalence of these definitions follows from a more general fact: Restriction of automorphism to subgroup invariant under it and its inverse is automorphism. In other words, we use the fact that both $\phi$ and ${\phi }^{-1}$ send $H$ to within itself to show that $\phi (H)=H$. It is not in general true that if an automorphism of a group restricts to a subgroup, then the restriction is an automorphism of the subgroup: Restriction of automorphism to subgroup not implies automorphism.

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.

Examples

Extreme examples

1. Every group is characteristic as a subgroup of itself.
2. The trivial subgroup is characteristic in any group.

Examples in Abelian groups

1. High occurrence example: In a cyclic group, every subgroup is characteristic. This can be seen, from instance, from the fact that every subgroup can be described as the set of all $d^{th}$ powers for some $d$, and the set of $d^{th}$ powers is invariant under automorphisms.
2. More generally, in an Abelian group, the set of $d^{th}$ powers, for any $d$, forms a characteristic subgroup (in fact, a fully characteristic subgroup, and even a verbal subgroup). Similarly, the set of elements whose order divides $d$, forms a characteristic subgroup (in fact, a fully characteristic subgroup).
3. Low occurrence example: In an elementary Abelian group, there are no characteristic subgroups other than the whole group and the trivial subgroup. This can be seen by viewing the elementary Abelian group as a vector space over a prime field, and observing that the automorphisms act transitively on the nonzero elements. Thus, no proper subgroup can be invariant under all automorphisms.
4. Low occurrence example: A group having no proper nontrivial characteristic subgroup is termed characteristically simple, and the above argument shows that, in general, a group whose automorphism group is transitive on non-identity elements, such as the additive group of a field or of a vector space over a field, is characteristically simple.

Examples in non-Abelian groups

1. In a non-Abelian group, some typical examples of characteristic subgroups are given by subgroup-defining functions (something which uniquely returns a particular subgroup). For instance, the Frattini subgroup, commutator subgroup, and center of any group, are characteristic. Similarly, all terms of the upper central series, lower central series, Frattini series, derived series, Fitting series and other series associated with the group, are characteristic.
2. For a finite group, any normal Sylow subgroup, and more generally, any normal Hall subgroup, is characteristic. More generally, the normal core of any Sylow subgroup or any Hall subgroup, is characteristic.

Some specific examples:

1. Example: In the symmetric group on three letters, the cyclic subgroup of order three (also known as the alternating group) is a normal Sylow subgroup, and hence characteristic. It is also the commutator subgroup of the big group.
2. Non-example: In the quaternion group, there are three cyclic subgroups of order four. All of them are normal, but none of them are characteristic (in fact, they are automorphs of each other). The intersection of all these three subgroups is a two-element subgroup that is characteristic: it equals the center, commutator as well as Frattini 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
View other second-order subgroup properties

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

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 $\to$ 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 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

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:

Characteristic = Automorphic subgroups $⟹$ 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.

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:

Stronger properties

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

Get a more complete list here

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

Get a more complete list here

Relation with normality

• Characteristic versus normal: Compares the subgroup properties of characteristicity and normality.
• Between normal and characteristic and beyond: A survey of the subgroup properties lying between normality and characteristicity. Get a list of all intermediate properties here.
• From normal to characteristic: A survey article on subgroup properties such that any normal subgroup satisfying that property is also characteristic.

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 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 transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

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 other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

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

Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this 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

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

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

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

For full proof, refer: Characteristicity is quotient-transitive

If $H_1$ is characteristic in $G_1$ and $H_2$ is characteristic in $G_2$, it is not necessary that $H_1\times H_2$ be characteristic in $G_1$ × $G_2$. A trivial counterexample is to set $G_1=G_2=G$ and $H_1=G$ and $H_2$ trivial. Here, the direct product is $G_1$ itself, but we know that the coordinate exchange automorphism takes $G_1$ to $G_2$, hence $G_1$ cannot be characteristic.

Intermediate subgroup condition

NO: This subgroup property does not satisfy the intermediate subgroup condition: it is possible to have a subgroup satisfying the property in the whole group but not satisfying the property in some 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 SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about intermediate subgroup condition

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

For full proof, refer: Characteristicity does not satisfy intermediate subgroup condition

Image condition

This subgroup property does not satisfy the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property need not satisfies the property

If $H$ is a characteristic subgroup of $G$, and $f:G\to K$ is a surjective homomorphism, then $f(H)$ need not be a characteristic subgroup of $K$. In fact, it need not be a characteristic subgroup even if the kernel of $f$ is also a characteristic subgroup.

For full proof, refer: Characteristicity does not satisfy image condition

Upper join-closedness

NO: This subgroup property is not upper join-closed: if a subgroup has the property in intermediate subgroups it need not have the property in their join.

If $H\le G$ and $K,L$ are intermediate subgroups such that $H$ is characteristic in both $K$ and $L$, $H$ need not be characteristic in the join $⟨K,L⟩$.

For full proof, refer: Characteristicity is not upper join-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 characteristic subgroups is characteristic. This follows from the more general fact that characteristicity is an endo-invariance property.

For full proof, refer: Characteristicity is commutator-closed

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 characteristic subgroup is characteristic. This follows from the more general fact that characteristicity can be expressed as an auto-invariance property: the invariance property corresponding to a collection of automorphisms. Any auto-invariance property is centralizer-closed.

For full proof, refer: Characteristicity is centralizer-closed

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

This subgroup property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
The GAP command for testing this subgroup property is:IsCharacteristicSubgroup
The GAP command for listing all subgroups with this property is:CharacteristicSubgroups
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 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.

The list of all characteristic subgroups can be obtained by:

CharacteristicSubgroups(group);

Study of the notion

Mathematical subject classification

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

References

Historical references

• (unknown) by Ferdinand Georg Frobenius, Berliner Sitzungsberichte, (Year 1895): ^{More info}, Page 183

Textbook references

• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, Page 17, (formal definition in paragraph)^{More info}
• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, Page 135, Section 4.4 (Automorphisms), (formal definition)^{More info}
• Topics in Algebra by I. N. Herstein, Page 70, Problem 7(a), (introduced in exercise)^{More info}
• Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, Page 234, Section 8 (generators and relations), Exercise 7, (introduced in exercise)^{More info}
• A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907, Page 428, Exercises 8.6, Concepts, Point 4, (introduced in exercise)^{More info}
• Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716, Page 168, ^{More info}
• Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, Page 4, Section 1.1, (definition in paragraph)^{More info} with a few important facts about characteristic and normal subgroups.

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