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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:

or equivalently, for all .

Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 History 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 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.[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 |
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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 SUBGROUP PROPERTY: Central factor: A subgroup with the property that every inner automorphism of the whole group restricts to an inner automorphism of the subgroup.

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.

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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
verbal subgroup	defined as the set of elements expressible by certain words	verbal implies fully invariant	fully invariant not implies verbal	click here
existentially bound-word subgroup	defined as the set of elements satisfying a system of equations	existentially bound-word implies fully invariant	fully invariant not implies existentially bound-word
homomorph-containing subgroup	contains every homomorphic image	homomorph-containing implies fully invariant	fully invariant not implies homomorph-containing	click here
subhomomorph-containing subgroup	contains every homomorphic image of every subgroup	(via homomorph-containing)	(via homomorph-containing)	click here
order-containing subgroup	contains every subgroup whose order divides its order	(via homomorph-containing)	(via homomorph-containing)	click here
variety-containing subgroup	contains every subgroup in the variety of groups generated by it	(via homomorph-containing subgroup)	(via homomorph-containing subgroup)	click here
normal subgroup having no nontrivial homomorphism to its quotient group	No nontrivial homomorphism to quotient group			click here
normal Hall subgroup	normal and Hall: its order and index are relatively prime			click here
normal Sylow subgroup	normal and Sylow			click here
quotient-subisomorph-containing subgroup		Quotient-subisomorph-containing implies fully invariant	Fully invariant not implies quotient-subisomorph-containing	click here
image-closed fully invariant subgroup	Under any surjective homomorphism, its image is fully invariant in the image of the group		full invariance does not satisfy image condition	click here
intermediately fully invariant subgroup	Fully invariant in every intermediate subgroup		full invariance does not satisfy intermediate subgroup condition
transfer-closed fully invariant subgroup	Its intersection with any subgroup is fully invariant in that		full invariance does not satisfy transfer condition

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions	Comparison
characteristic subgroup	invariant under all automorphisms	fully invariant implies characteristic	characteristic not implies fully invariant (see also list of examples)	click here	characteristic versus fully invariant
normal subgroup	invariant under all inner automorphisms	(via characteristic)	(via characteristic)	click here
strictly characteristic subgroup	invariant under all surjective endomorphisms	fully invariant implies strictly characteristic	strictly characteristic not implies fully invariant	click here	--
injective endomorphism-invariant subgroup	invariant under all injective endomorphisms		injective endomorphism-invariant not implies fully invariant
retraction-invariant subgroup	invariant under all retractions			click here
retraction-invariant characteristic subgroup	characteristic and retraction-invariant
retraction-invariant normal subgroup	normal and retraction-invariant
endomorph-dominating subgroup	every image under an endomorphism is conjugate to a subgroup of it
potentially fully invariant subgroup	the subgroup is fully invariant in some bigger group			click here
finite direct power-closed characteristic subgroup	any irect power of the subgroup is characteristic in the corresponding direct power of the whole group	follows from full invariance is finite direct power-closed and fully invariant implies characteristic	finite direct power-closed characteristic not implies fully invariant

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes	full invariance is transitive	If , with H fully invariant in K and K fully invariant in G, then H is fully invariant in G.
trim subgroup property	Yes		The trivial subgroup and the whole group are always fully invariant.
intermediate subgroup condition	No	full invariance does not satisfy intermediate subgroup condition	It is possible to have such that H is a fully invariant subgroup inside G but is not a fully invariant subgroup of K.
strongly intersection-closed subgroup property	Yes	full invariance is strongly intersection-closed	If , are all fully invariant subgroups of G, then is also fully invariant in G.
strongly join-closed subgroup property	Yes	full invariance is strongly join-closed	If , are all fully invariant subgroups of G, then is also fully invariant in G.
commutator-closed subgroup property	Yes	full invariance is commutator-closed	If H,K are fully invariant subgroups of G, so is .
quotient-transitive subgroup property	Yes	full invariance is quotient-transitive	If such that H is fully invariant in G and K / H is fully invariant in G / H, then K is fully invariant in G.
finite direct power-closed subgroup property	Yes	full invariance is finite direct power-closed	If H is fully invariant in G, then in any finite direct power Gn of G, the corresponding direct power Hn is fully invariant.
restricted direct power-closed subgroup property	Yes	full invariance is restricted direct power-closed	If H is fully invariant in G, then in any restricted direct power of G, the corresponding direct power of H is fully invariant.

Here is more information on these metaproperties: [SHOW MORE]

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 command| View subgroup properties for which all subgroups can be listed with built-in GAP commands |
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

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Definition links

• Wikipedia page
• Planetmath page