Visit

Groupprops, The Group Properties Wiki (pre-alpha)

TIP: Having trouble locating the wiki page for a term/definition? Get tips

ABOUT US: We use a Creative Commons license. All our content is free to reuse, with attribution. Learn more

ALSO CHECK OUT: Topospaces: The Topology Wiki

From Groupprops

This article is about a standard (though not very rudimentary) definition in group theory.[SHOW MORE]

The article text may, however, contain more than just the basic definition
VIEW: Definitions built on this | Facts about this | Survey articles about this | Facts about definitions built on this |
VIEW RELATED: Analogues of this | Variations of 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[SHOW MORE]

VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions | Subgroup property dissatisfactions
VIEW FACTS THAT: use, or start with, a subgroup satisfying the property| prove, or show that, a subgroup satisfies the property
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
Find other variations of characteristicity | 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

QUICK PHRASES: invariant under all endomorphisms, endomorphism-invariant

Symbol-free definition

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

Definition with symbols

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

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

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

Examples

1. High occurrence example: In a cyclic group, every subgroup is fully invariant. 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 invariant. 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 invariant 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 invariant 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 invariant. Further information: center not is fully invariant

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 invariant 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 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 fully invariant subgroup of G if ...	This means that full invariance is ...	Additional comments
endomorphism function	every endomorphism of G sends every element of H to within H	the invariance property for endomorphisms
endomorphism endomorphism	every endomorphism of G restricts to an endomorphism of H	the balanced subgroup property for endomorphisms	Hence, it is a t.i. subgroup property, both transitive and identity-true
endomorphism endomorphism	every endomorphism of G restricts to an endomorphism of H	the endo-invariance property for endomorphisms; i.e., it is the invariance property for endomorphism, which is a property stronger than the property of being an endomorphism

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

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 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 being fully invariant is a transitive subgroup property. That is, a fully invariant subgroup of a fully invariant subgroup is fully invariant. This is because full invariance is a balanced subgroup property in the function restriction formalism.

For full proof, refer: Full invariance is transitive

Further information: Balanced implies transitive, 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 property of being fully invariant is trim, in the sense:

• The trivial subgroup is always fully invariant.
• The whole group is always fully invariant in itself.

Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
In fact, since the property is also true for every group as a subgroup of itself, it is a strongly intersection-closed subgroup property.
ABOUT THIS PROPERTY: |
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

The property of being fully invariant is intersection-closed. That is, the intersection of a family of fully invariant subgroups is fully invariant. This follows from the fact that the property of being fully invariant is an invariance property.

For full proof, refer: Full invariance is strongly intersection-closed

Further information: Invariance implies 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.
In fact, since the property is also true for the trivial subgroup in any group, it is a strongly join-closed subgroup property.
ABOUT THIS PROPERTY: |
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

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

For full proof, refer: Full invariance is strongly join-closed

Further information: Endo-invariance implies strongly 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

If are fully invariant subgroups, the commutator [H,K] is also fully invariant.

For full proof, refer: Full invariance is commutator-closed

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties

If are such that H is fully invariant in G and K / H is fully invariant in G / H, then K is fully invariant in G.

For full proof, refer: Full invariance is quotient-transitive

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

Definition links

• Wikipedia page
• Planetmath page