Characteristic subgroup
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Definition
QUICK PHRASES: invariant under all automorphisms, automorphism-invariant, strongly normal, normal under outer automorphisms
Equivalent definitions in tabular format
Below are many equivalent definitions of characteristic subgroup.
No. | Shorthand | A subgroup of a group is characteristic in it if ... | A subgroup of a group is called a characteristic subgroup of if ... |
---|---|---|---|
1 | automorphism-invariant | every automorphism of the whole group takes the subgroup to within itself. | for every automorphism of , . More explicitly, for any and , . |
2 | automorphisms restrict to endomorphisms | every automorphism of the group restricts to an endomorphism of the subgroup. | for every automorphism of , and restricts to an endomorphism of . |
3 | automorphisms restrict to automorphisms | every automorphism of the group restricts to an automorphism of the subgroup. | for every automorphism of , and restricts to an automorphism of . |
4 | (definition in terms of Hom-set maps) | (too complicated to state without symbols) | the image in under the natural map (by restriction) is contained in the image in under the natural map (by inclusion). |
5 | left transiter of normality | it is normal in any supergroup in which the whole group is normal. | for any group containing as a normal subgroup, is also normal in . |
This definition is presented using a tabular format. |View all pages with definitions in tabular format
Notation and terminology
For a subgroup of a group , we denote the characteristicity of in by .
Equivalence of definitions
The equivalence of definitions (1) to (4) 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 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.
For the equivalence with definition (5), see left transiter of normal is characteristic.
Copyable LaTeX
A subgroup $H$ of a group $G$ is termed a {\em characteristic subgroup} if $\varphi(H) = H$ for all automorphisms $\varphi$ of $G$.
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: (facts closely related to Characteristic subgroup, all facts related to Characteristic subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites 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
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
Type of group | High occurrence or low occurrence? | Some or all characteristic subgroups | Explanation/comment |
---|---|---|---|
cyclic group | high | every subgroup is characteristic: cyclic implies every subgroup is characteristic | Every subgroup can be described as the set of powers for some . This set is invariant under automorphisms. |
abelian group | -- | the set of powers, i.e., elements of the form for fixed integer , variable (in additive notation, this becomes ) | This set is not just a characteristic subgroup, it is in fact a fully invariant subgroup and a verbal subgroup. See power subgroup. |
elementary abelian group, more generally additive group of a field | low | no characteristic subgroups other than trivial subgroup and whole group | The additive group of a field or of a vector space over a field is a group whose automorphism group is transitive on non-identity elements. Hence, it is a characteristically simple group -- any nontrivial characteristic subgroup must be the whole group. |
Examples using subgroup-defining functions across all 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, we have the following subgroup-defining functions, some of which are interesting for abelian groups as well:
Subgroup-defining function | Definition | Proof of characteristicity |
---|---|---|
Frattini subgroup | intersection of all maximal subgroups | Frattini subgroup is characteristic |
derived subgroup | subgroup generated by all commutators | derived subgroup is characteristic |
center | set of elements that commute with every element | center is characteristic |
For a complete list of subgroup-defining functions, see Category:Subgroup-defining functions.
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.
Examples of subgroups satisfying the property
Here are some examples of subgroups in basic/important groups satisfying the property:
Group part | Subgroup part | Quotient part | |
---|---|---|---|
A3 in S3 | Symmetric group:S3 | Cyclic group:Z3 | Cyclic group:Z2 |
Here are some examples of subgroups in relatively less basic/important groups satisfying the property:
Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:
Examples of subgroups not satisfying the property
Here are some examples of subgroups in basic/important groups not satisfying the property:
Group part | Subgroup part | Quotient part | |
---|---|---|---|
S2 in S3 | Symmetric group:S3 | Cyclic group:Z2 | |
Z2 in V4 | Klein four-group | Cyclic group:Z2 | Cyclic group:Z2 |
Here are some some examples of subgroups in relatively less basic/important groups not satisfying the property:
Here are some examples of subgroups in even more complicated/less basic groups not satisfying the property:
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 | Difficulty level (0-5) | Statement with symbols |
---|---|---|---|---|
transitive subgroup property | Yes | characteristicity is transitive | 1 | If are groups such that is characteristic in and is characteristic in , then is characteristic in . |
trim subgroup property | Yes | obvious reasons | 0 | and are characteristic in |
strongly intersection-closed subgroup property | Yes | characteristicity is strongly intersection-closed | 1 | If , are all characteristic in , so is the intersection of subgroups . |
strongly join-closed subgroup property | Yes | characteristicity is strongly join-closed | 1 | If , are all characteristic in , so is the join of subgroups . |
quotient-transitive subgroup property | Yes | characteristicity is quotient-transitive | 1 | If , with characteristic in and characteristic in , then is characteristic in . |
intermediate subgroup condition | No | characteristicity does not satisfy intermediate subgroup condition | 2 | We can have such that is characteristic in but is not characteristic in . |
upper join-closed subgroup property | No | characteristicity is not upper join-closed | 3 | We can have and intermediate subgroups such that is characteristic in both and but not in . |
commutator-closed subgroup property | Yes | characteristicity is commutator-closed | 1 | If are characteristic in , so is . |
centralizer-closed subgroup property | Yes | characteristicity is centralizer-closed | 1 | If is characteristic in , so is . |
lower central series condition | No | characteristicity does not satisfy lower central series condition | (second member case): It is possible to have a group and a characteristic subgroup such that the derived subgroup is not characteristic inside the derived subgroup . | |
finite direct power-closed subgroup property | No | characteristicity is not finite direct power-closed | 3 | ( case): It is possible to have a group and a characteristic subgroup such that is not a characteristic subgroup of . |
finite-relative-intersection-closed subgroup property | No | characteristicity is not finite-relative-intersection-closed | 2 | It is possible to have a group with subgroups , such that , characteristic in and characteristic in , but not characteristic in . |
partition difference condition | Yes | characteristicity satisfies partition difference condition | 1 | Suppose is a group and is a subgroup of with a partition into two or more subgroups . Then, if all except possibly one of the s are characteristic subgroups of , all the s are characteristic subgroups of . |
conditionally lattice-determined subgroup property | No | No subgroup property between normal Sylow and subnormal or between Sylow retract and retract is conditionally lattice-determined | It is possible to have a group , an automorphism of the lattice of subgroups, and a normal Sylow subgroup of such that is not subnormal. In particular, is characteristic in and is not. |
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).
For a complete list of subgroup properties stronger than Characteristic subgroup, click here
STRONGER PROPERTIES SATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | trim | inverse image condition | image condition | centralizer-closed |
STRONGER PROPERTIES DISSATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | trim | inverse image condition | image condition | centralizer-closed |
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):
View a complete list of conjunctions of characteristicity with subgroup properties
Here are important conjunctions of the property of being a characteristic subgroup with group properties:
View a complete list of conjunctions of characteristicity with group properties
In some cases, we are interested in studying characteristic subgroups where the big group is constrained to satisfy some group property. For instance:
Weaker properties
For a complete list of subgroup properties weaker than Characteristic subgroup, click here
WEAKER PROPERTIES SATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | trim | inverse image condition | image condition | centralizer-closed |
WEAKER PROPERTIES DISSATISFYING SPECIFIC METAPROPERTIES: transitive | intermediate subgroup condition | transfer condition | quotient-transitive |intersection-closed |join-closed | trim | inverse image condition | image condition | centralizer-closed |
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.
Analogues in other algebraic structures
Algebraic structure | Analogue of characteristic subgroup in that structure | Definition | Nature of analogy with characteristic subgroup |
---|---|---|---|
Lie ring | characteristic Lie subring | A subring of a Lie ring that is invariant under all automorphisms of the whole Lie ring. | Same idea of subalgebra and automorphisms, but for Lie ring structure instead of group structure. |
Lie ring | characteristic ideal of a Lie ring | A subring of a Lie ring that arises as the kernel of a homomorphism and is invariant under all automorphisms | Note that since characteristic implies normal, the characteristic subgroups coincide with the characteristic normal subgroups |
variety of algebras (universal algebra) | characteristic subalgebra | subalgebra that is invariant under all automorphisms | same definition, applied to an arbitrary algebraic structure. |
loop | characteristic subloop | subloop that is invariant under all automorphisms | same definition, applied to another algebraic stricture. |
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 | normal equals potentially characteristic |
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 | normal equals image-potentially characteristic |
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 |
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)
Varietal formalism
The notion of characteristic subgroup can be defined as the notion of characteristic subalgebra in the variety of groups.
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 is characteristic in 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 | is a characteristic subgroup of if ... | This means that characteristicity is ... | Additional comments |
---|---|---|---|
automorphism function | every automorphism of sends every element of to within | the invariance property for automorphisms | |
automorphism endomorphism | every automorphism of restricts to an endomorphism of | 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 restricts to a automorphism of | 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 (i.e., 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.
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);
State of discourse
Mathematical subject classification
Under the Mathematical subject classification, the study of this notion comes under the class: 20A05
Broadly speaking, "characteristic subgroups" are not a separate area of study in group theory, but rather, characteristic subgroups are an important part of the vocabulary that comes up in any sufficiently "pure" study of group theory. Characteristic subgroups are relatively less important in the study of group actions, and more important in the study of abstract group structure.
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.
Resolution of questions that are easy to formulate
Any typical question about the behavior of characteristic subgroups in arbitrary groups that is easy to formulate will also be easy to resolve either with a proof or a counterexample, unless some other feature of the question significantly complicates it. This is so, despite the fact that there are a large number of easy-to-formulate questions about the automorphism group that are still open. The reason is that even though not enough is known about the automorphism groups, there are other ways to obtain information about the structure of characteristic subgroups.
At the one extreme, there are abelian groups, where the characteristic subgroups are quite easy to get a handle on. At the other extreme, there are "all groups" where very little can be said about characteristic subgroups beyond what can be proved through elementary reasoning. The most interesting situation is in the middle, for instance, when we are looking at nilpotent groups and solvable groups. In these cases, there are some restrictions on the structure of characteristic subgroups, but the exact nature of the restrictions is hard to work out.
Below are listed some of the more cutting-edge questions that are relatively easy to formulate and are partially open.
Problem description | More details |
---|---|
potentially characteristic subgroups characterization problem | This problem (or rather, constellation of problems) asks, given a subgroup , whether there exists a group containing such that is characteristic in . Questions of this type are generally much harder to resolve than formulate. |
characteristic not implies powering-invariant in nilpotent group | This disproves the conjecture that if is a characteristic subgroup of a nilpotent group , then is a powering-invariant subgroup of , i.e., if is powered over a prime (unique -divisibility), so is . |
Bryant-Kovacs theorem | This theorem provides a "rigidification" result that guarantees the existence of finite p-groups with specific patterns of characteristic subgroups containing the Frattini subgroup. |
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, published 1911, 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 978-0471433347^{More info} | 135 | Section 4.4 (Automorphisms) | formal definition | |
Topics in Algebra by I. N. Herstein^{More info} | 70 | Problem 7(a) | introduced in exercise | |
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632^{More 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 0201763907^{More info} | 428 | Exercises 8.6, Concepts, Point 4 | introduced in exercise | |
Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716^{More info} | 168 | |||
Basic Algebra: Groups, Rings, and Fields by Paul Moritz Cohn, ISBN 1852335874, 13-digit ISBN 978-1852335878^{More info} | 46 | Section 2.6 | definition in paragraph tangential to the topic of discussion. | Google Books |
Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189^{More info} | 103 | Chapter II, Section 7 | definition in paragraph precursor to results using the idea. | 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 978-0821843444^{More info} | 11 | definition in paragraph | Google Books | |
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261^{More info} | 17 | formal definition in paragraph | Google Books | |
Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724^{More 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 0521786754^{More info} | 25 | Google Books | ||
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More 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 978-0387942858^{More 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)