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Fully invariant subgroup

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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 \varphi of G:

\varphi(H) \le H

or equivalently, \varphi(x) \in H for all x \in H.


Contents

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

\forall \ g \in H, \forall \sigma \in G^G : \ (\ \forall \ a,b \in G, \sigma(ab) = \sigma(a)\sigma(b)) \implies \sigma(g) \in H

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 \to function every endomorphism of G sends every element of H to within H the invariance property for endomorphisms
endomorphism \to 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 \to 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 invariantFully 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 H \le K \le G, 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 H \le K \le G 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 H_i, i \in I, are all fully invariant subgroups of G, then \bigcap_{i \in I} H_i is also fully invariant in G.
strongly join-closed subgroup property Yes full invariance is strongly join-closed If H_i, i \in I, are all fully invariant subgroups of G, then \langle H_i \rangle_{i\in I} 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 \! [H,K].
quotient-transitive subgroup property Yes full invariance is quotient-transitive If H \le K \le G 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

More info

Textbook references

External links

Search for "fully+characteristic+subgroup"OR"fully+invariant+subgroup" on the World Wide Web:
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Encyclopaedias: Wikipedia (or using Google), Citizendium
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Definition links

Facts about Fully invariant subgroupRDF feed
Defined inPaper:Levi33 (?, ?, ?)  +, Paper:Baer44 (?, ?, ?)  +, and Book:RobinsonGT (?, ?, ?)  +
Dissatisfies metapropertyIntermediate subgroup condition  +, Transfer condition  +, and Inverse image condition  +
Left side of function restriction expressionEndomorphism  +
Page classTerm  +
Quick phraseinvariant under all endomorphisms  +, and endomorphism-invariant  +
Referenced inPaper:Levi33 (?, ?, ?)  +, Paper:Baer44 (?, ?, ?)  +, Book:RobinsonGT (?, ?, ?)  +, Wikipedia (?, ?, ?)  +, and planetmath (?, ?, ?)  +
Right side of function restriction expressionEndomorphism  +
Satisfies metapropertyTransitive subgroup property  +, Trim subgroup property  +, Trivially true subgroup property  +, Identity-true subgroup property  +, Left-realized subgroup property  +, Right-realized subgroup property  +, Intersection-closed subgroup property  +, Strongly intersection-closed subgroup property  +, Join-closed subgroup property  +, Strongly join-closed subgroup property  +, Commutator-closed subgroup property  +, Quotient-transitive subgroup property  +, Second-order subgroup property  +, Function restriction-expressible subgroup property  +, Invariance property  +, Balanced subgroup property  +, Endo-invariance property  +, Finite direct power-closed subgroup property  +, Restricted direct power-closed subgroup property  +, and GAP-testable subgroup property  +
Stronger thanCharacteristic subgroup  +, Normal subgroup  +, Strictly characteristic subgroup  +, Injective endomorphism-invariant subgroup  +, Retraction-invariant subgroup  +, Retraction-invariant characteristic subgroup  +, Retraction-invariant normal subgroup  +, Endomorph-dominating subgroup  +, Potentially fully invariant subgroup  +, and Finite direct power-closed characteristic subgroup  +
Term introduced byLevi  +
Variation ofCharacteristicity  +
Weaker thanVerbal subgroup  +, Existentially bound-word subgroup  +, Homomorph-containing subgroup  +, Subhomomorph-containing subgroup  +, Order-containing subgroup  +, Variety-containing subgroup  +, Normal subgroup having no nontrivial homomorphism to its quotient group  +, Normal Hall subgroup  +, Normal Sylow subgroup  +, Quotient-subisomorph-containing subgroup  +, Image-closed fully invariant subgroup  +, Intermediately fully invariant subgroup  +, and Transfer-closed fully invariant subgroup  +
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