Characteristic subgroup
From Groupprops
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
of G,
. More explicitly, for any
and
,
- For every automorphism
of G,
and
restricts to an endomorphism of H.
- For every automorphism
of G,
and
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
and
send H to within itself to show that
. 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.
|
This article is about a basic definition in group theory.The article text may, however, contain advanced material.
VIEW: Definitions built on this | Facts about this | Survey articles about this | Facts about definitions built on this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | [SHOW MORE]
This article defines a subgroup property that is pivotal (viz important) among existing subgroup properties
View a list of pivotal subgroup properties | View a complete list of subgroup properties[SHOW MORE]
This is a variation of normality
Find other variations of normality | Read a survey article on varying normality
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.
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
VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions
Extreme examples
- Every group is characteristic as a subgroup of itself.
- The trivial subgroup is characteristic in any group.
Examples in abelian groups
- 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 dth powers for some d, and the set of dth powers is invariant under automorphisms.
- More generally, in an abelian group, the set of dth powers, for any d, forms a characteristic subgroup (in fact, a fully invariant subgroup, and even a verbal subgroup). Similarly, the set of elements whose order divides d, forms a characteristic subgroup (in fact, a fully invariant subgroup).
- 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.
- 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
- 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.
- 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:
- 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.
- 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
The second-order description of characteristicity is as follows. We say H is characteristic in G if:
The key point is that quantification over
is a second-order quantification.
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 characteristic subgroup of G if ... | This means that characteristicity is ... | Additional comments |
|---|---|---|---|
automorphism function | every automorphism of G sends every element of H to within H | the invariance property for automorphisms | |
automorphism endomorphism | every automorphism of G restricts to an endomorphism of H | the endo-invariance property for automorphisms; i.e., it is the invariance property for automorphism, which is a property stronger than the property of being an endomorphism | |
automorphism automorphism | every automorphism of G restricts to a automorphism of H | the balanced subgroup property for automorphisms | Hence, it is a t.i. subgroup property, both transitive and identity-true |
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:
- Variations of characteristic subgroup
- Analogues of characteristic subgroup in other algebraic structures
Stronger properties
The most important stronger properties are fully invariant subgroup (invariant under all endomorphisms) and isomorph-free subgroup (no other isomorphic subgroup).
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions | Collapse |
|---|---|---|---|---|---|
| fully invariant subgroup (also called fully characteristic subgroup) | invariant under all endomorphisms | fully invariant implies characteristic | characteristic not implies fully invariant (see also list of examples) | click here | group in which every characteristic subgroup is fully invariant |
| isomorph-containing subgroup | contains all isomorphic subgroups | isomorph-containing implies characteristic | characteristic not implies isomorph-containing (see also list of examples) | click here | group in which every characteristic subgroup is isomorph-containing |
| isomorph-free subgroup | no other isomorphic subgroups | (via isomorph-containing) | (via isomorph-containing) (see also list of examples) | click here | group in which every characteristic subgroup is isomorph-free |
Conjunction with other properties
Important conjunctions of characteristicity with other subgroup properties (Note that multiple properties listed in the second column indicate that any one of them can be used):[SHOW MORE]Here are important conjunctions of the property of being a characteristic subgroup with group properties:[SHOW MORE]
In some cases, we are interested in studying characteristic subgroups where the big group is constrained to satisfy some group property. For instance:[SHOW MORE]
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions | Comparison | Collapse |
|---|---|---|---|---|---|---|
| normal subgroup | invariant under inner automorphisms | characteristic implies normal | normal not implies characteristic (see also list of examples) | click here | normal versus characteristic | group in which every normal subgroup is characteristic |
| subnormal subgroup | chain from subgroup to group, each normal in next | (via normal) | (via normal) | click here | -- | group in which every normal subgroup is characteristic |
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.
- Subnormal-to-normal and 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)
Here is a summary:
| Metaproperty name | Satisfied? | Proof | Statement with symbols |
|---|---|---|---|
| Transitive subgroup property | Yes | Characteristicity is transitive | If H is characteristic in K and K is characteristic in G, then H is characteristic in G |
| Trim subgroup property | Yes | Obvious reasons | {e} and G are characteristic in G |
| Strongly intersection-closed subgroup property | Yes | Characteristicity is strongly intersection-closed | If , are all characteristic in G, so is .
|
| Strongly join-closed subgroup property | Yes | Characteristicity is strongly join-closed | If , are all characteristic in G, so is
|
| Quotient-transitive subgroup property | Yes | Characteristicity is quotient-transitive | If , with H characteristic in G and K / H characteristic in G / H, then K is characteristic in G
|
| Intermediate subgroup condition | No | Characteristicity does not satisfy intermediate subgroup condition | We can have with H characteristic in G but not in K
|
| Upper join-closed subgroup property | No | Characteristicity is not upper join-closed | We can have and K1,K2 intermediate subgroups such that H is characteristic in both but not in .
|
| Commutator-closed subgroup property | Yes | Characteristicity is commutator-closed | If H,K are characteristic in G, so is [H,K] |
| Centralizer-closed subgroup property | Yes | Characteristicity is centralizer-closed | If H is characteristic in G, so is CG(H) |
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)
| Operator | Meaning | Result of application | Proof |
|---|---|---|---|
| potentially operator | characteristic in some larger group | normal subgroup | NPC theorem |
| simple group operator | no proper nontrivial characteristic subgroup | characteristically simple group | |
| image-potentially operator | exists as the image of a characteristic subgroup via a surjective homomorphism | normal subgroup | NIPC theorem |
| intermediately operator | characteristic in every intermediate group | intermediately characteristic subgroup | |
| transfer condition operator | intersection with every subgroup is characteristic in it | transfer-closed characteristic subgroup | |
| image condition operator | image for every surjective homomorphism is characteristic in the image | image-closed characteristic subgroup |
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 |
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
- Theory of Groups of Finite Order by William Burnside, Page 92, More info: The terminology and language in this book is antiquated, and is only of historical interest. View on Google Books
Textbook references
Advanced undergraduate/beginning graduate algebra texts:
| Book | Page number | Chapter and section | Contextual information | View |
|---|---|---|---|---|
| Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347More info | 135 | Section 4.4 (Automorphisms) | formal definition | |
| Topics in Algebra by I. N. HersteinMore info | 70 | Problem 7(a) | introduced in exercise | |
| Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632More info | 234 | Section 8 (generators and relations), Exercise 7 | introduced in exercise | |
| A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907More info | 428 | Exercises 8.6, Concepts, Point 4 | introduced in exercise | |
| Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716More info | 168 | |||
| Basic Algebra: Groups, Rings, and Fields by Paul Moritz Cohn, ISBN 1852335874, 13-digit ISBN 978-1852335878More info | 46 | Section 2.6 | definition in paragraph tangential to the topic of discussion. | Google Books |
Graduate texts on group theory:
| Book | Page number | Chapter and section | Contextual information | View |
|---|---|---|---|---|
| Finite Group Theory by I. Martin Isaacs, ISBN 0821843443, 13-digit ISBN 978-0821843444More info | 11 | definition in paragraph | Google Books | |
| Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261More info | 17 | formal definition in paragraph | Google Books | |
| Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724More info | 4 | Section 1.1 | definition in paragraph with a few important facts about characteristic and normal subgroups. | |
| Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754More info | 25 | Google Books | ||
| A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613More info | 28 | Section 1.5 | definition in paragraph | Google Books |
| An introduction to the theory of groups by Joseph J. Rotman, ISBN 0387942858, 13-digit ISBN 978-0387942858More info | 104 | formal definition | Google Books |
Online lecture notes
- J.S. Milne's course notes, Section 3.2, Page 41 (both the A4 and the letter version)
External links
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Definition links
Names in other languages:German: Charakteristische Untergruppe; French: Sous-groupe caractéristique; Italian: Sottogruppo caratteristicoUse Google translate to translate this page to French, German, Spanish, Italian
function
, are all characteristic in
.
, with
and
.
be characteristic in
is a surjective homomorphism, then
.