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Contents 1 History 2 Definition 2.1 Symbol-free definition 2.2 Definition with symbols 3 Examples 3.1 Extreme examples 3.2 Examples 3.3 Non-examples 4 Formalisms 4.1 Second-order description 4.2 Function restriction expression 5 Relation with other properties 5.1 Stronger properties 5.2 Weaker properties 6 Metaproperties 6.1 Transitivity 6.2 Trimness 6.3 Intersection-closedness 6.4 Join-closedness 7 Testing 7.1 GAP command 8 References 8.1 Journal references 8.2 Textbook references 9 External links 9.1 Definition links

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a complete list of semi-basic definitions on this wiki

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
View a complete list of subgroup properties|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions | Subgroup property dissatisfactions
RANDOM TIP:The testing section provides information on practical testing for the subgroup property, including implementation in GAP, when possible.

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

History

This term was introduced by: Levi

The concept was introduced by Levi in 1933 under the German name vollinvariant (translating to fully invariant). Both the terms fully characteristic and fully invariant are now in vogue.

Definition

Symbol-free definition

A subgroup of a group is termed fully characteristic or fully invariant if it is invariant under all endomorphisms of the whole group.

Definition with symbols

A subgroup H of a group G is termed fully characteristic if, for any endomorphism of G:

Examples

Extreme examples

1. The trivial subgroup is always fully characteristic.
2. Every group is fully characteristic as a subgroup of itself.

Examples

1. High occurrence example: In a cyclic group, every subgroup is fully characteristic. That's because any subgroup can be described as the set of all d^th powers, for some choice of d, and such a set is clearly invariant under endomorphisms. (In fact, it is a verbal subgroup).
2. More generally, in any Abelian group, the set of d^th powers is a verbal subgroup, and hence fully characteristic. The set of elements whose order divides d is also fully characteristic, though not necessarily verbal (for instance, in the group of all roots of unity, the subgroup of n^th roots for fixed n is fully characteristic but not verbal).
3. In a (possibly) non-Abelian group, certain subgroup-defining functions always yield a fully characteristic subgroup. For instance, the commutator subgroup is fully characteristic, and so are all terms of the lower central series as well as the derived series.

Non-examples

1. In an elementary Abelian group, and more generally, in a characteristically simple group, there is no proper nontrivial fully characteristic subgroup (in fact, there's no proper nontrivial characteristic subgroup, either).
2. There do exist characteristic subgroups that are not fully characteristic; in fact, the center, and terms of the upper central series, may be characteristic but not fully characteristic. Further information: center not is fully characteristic

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 property of being fully characteristic has a second-order description. A subgroup H of a group G is termed fully characteristic if:

The condition in parentheses is a verification that the function σ is an endomorphism of G.

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

The property of being fully characteristic can be expressed in terms of the function restriction formalism in the following ways:

• As the invariance property with respect to the function property of being an endomorphism:

Fully characteristic = Endomorphism Function

In other words, a subgroup is fully characteristic if and only if every endomorphism of the whole group restricts to a function on the subgroup (that is, takes the subgroup to within itself).

• As the balanced subgroup property (function restriction formalism) with respect to the function property of being an endomorphism:

Fully characteristic = Endomorphism Endomorphism

In other words, a subgroup is fully characteristic if and only if every endomorphism of the whole group restricts to an endomorphism of the subgroup.

Relation with other properties

Stronger properties

• Verbal subgroup: For full proof, refer: verbal implies fully characteristic
• Existentially bound-word subgroup: For full proof, refer: existentially bound-word implies fully characteristic
• Homomorph-containing subgroup
• Image-closed fully characteristic subgroup
• Intermediately fully characteristic subgroup

Weaker properties

• Strictly characteristic subgroup
• Retraction-invariant subgroup
• Characteristic subgroup
• Normal subgroup

Metaproperties

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

The property of being fully characteristic is a transitive subgroup property. That is, a fully characteristic subgroup of a fully characteristic subgroup is fully characteristic. This is because full characteristicity is a balanced subgroup property in the function restriction formalism. For full proof, refer: Full 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 property of being fully characteristic is trim, in the sense:

• The trivial subgroup is always fully characteristic
• The whole group is always fully characteristic in itself

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

The property of being fully characteristic is intersection-closed. That is, the intersection of a family of fully characteristic subgroups is fully characteristic. This follows from the fact that the property of being fully characteristic is an invariance property.For full proof, refer: Full characteristicity 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

The property of being fully characteristic is join-closed, viz an arbitrary join (subgroup generated) of fully characteristic subgroups is fully characteristic. This follows from the fact that full characteristicity is an endo-invariance property (an invariance property with respect to certain kinds of endomorphisms).

For full proof, refer: Full characteristicity is join-closed

Testing

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:IsFullinvariant
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

Note that this GAP testing function uses an additional package called the SONATA package.

References

Journal references

• Über die Untergruppen der freien Gruppen by Levi, Math. Zeit. vol. 37 (1933) pp. 90-9 (German): In this paper, Levi introduces, among other things, the concept of a fully characteristic subgroup (under the name vollinvariant, that translates to fully invariant).^{More info}
• The higher commutator subgroups of a group by Reinhold Baer , Bulletin of the American Mathematical Society, ISSN 10889485 (electronic), ISSN 02730979 (print), Page 143 - 160 (Year 1944): This paper compares invariance properties such as normal subgroup, characteristic subgroup, strictly characteristic subgroup, and fully characteristic subgroup (here, called fully invariant subgroup).^{Full text (PDF)}

^{More info}

Textbook references

• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 28, Characteristic and fully invariant subgroups

External links

Search for "fully+characteristic+subgroup"OR"fully+invariant+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
The web: Google, Yahoo, Windows Live
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Definition links

• Wikipedia page
• Planetmath page