Fully invariant subgroup

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Definition

QUICK PHRASES: invariant under all endomorphisms, endomorphism-invariant

Equivalent definitions in tabular format

A subgroup of a group is termed fully invariant or fully characteristic if it satisfies the following equivalent conditions:

No. Shorthand A subgroup of a group is termed fully invariant if ... A subgroup H of a group G is termed a fully invariant subgroup of G if ...
1 endomorphism-invariant it is invariant under all endomorphisms of the whole group. for any endomorphism \varphi of G, \varphi(H) \subseteq H or equivalently, \varphi(x) \in H for all x \in H.
2 endomorphism restricts to endomorphism every endomorphism of the whole group restricts to an endomorphism of the group. for any endomorphism \varphi of G, \varphi(H) \subseteq H and the restriction of \varphi to H is an endomorphism of H.


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
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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity


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 invariant, 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 invariant but not verbal).
  3. In a (possibly) non-abelian group, certain subgroup-defining functions always yield a fully invariant subgroup. For instance, the derived subgroup is fully invariant, 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 invariant; 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


Examples of subgroups satisfying the property

Here are some examples of subgroups in basic/important groups satisfying the property:

 Group partSubgroup partQuotient part
A3 in S3Symmetric group:S3Cyclic group:Z3Cyclic group:Z2

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

 Group partSubgroup partQuotient part
A4 in S4Symmetric group:S4Alternating group:A4Cyclic group:Z2
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
Center of quaternion groupQuaternion groupCyclic group:Z2Klein four-group
Center of special linear group:SL(2,3)Special linear group:SL(2,3)Cyclic group:Z2Alternating group:A4
Center of special linear group:SL(2,5)Special linear group:SL(2,5)Cyclic group:Z2Alternating group:A5
First agemo subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Klein four-group
First omega subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Klein four-groupCyclic group:Z2
Klein four-subgroup of alternating group:A4Alternating group:A4Klein four-groupCyclic group:Z3
Normal Klein four-subgroup of symmetric group:S4Symmetric group:S4Klein four-groupSymmetric group:S3

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

 Group partSubgroup partQuotient part
Center of M16M16Cyclic group:Z4Klein four-group
Center of dihedral group:D16Dihedral group:D16Cyclic group:Z2Dihedral group:D8
Center of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Klein four-groupKlein four-group
Center of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z2Dihedral group:D8
D8 in SD16Semidihedral group:SD16Dihedral group:D8Cyclic group:Z2
Derived subgroup of M16M16Cyclic group:Z2Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z4Klein four-group
Derived subgroup of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Direct product of Z4 and Z2
Direct product of Z4 and Z2 in M16M16Direct product of Z4 and Z2Cyclic group:Z2
Klein four-subgroup of M16M16Klein four-groupCyclic group:Z4
SL(2,3) in GL(2,3)General linear group:GL(2,3)Special linear group:SL(2,3)Cyclic group:Z2

Examples of subgroups not satisfying the property

Here are some examples of subgroups in basic/important groups not satisfying the property:

 Group partSubgroup partQuotient part
S2 in S3Symmetric group:S3Cyclic group:Z2
Z2 in V4Klein four-groupCyclic group:Z2Cyclic group:Z2

Here are some some examples of subgroups in relatively less basic/important groups not satisfying the property:

 Group partSubgroup partQuotient part
A3 in A4Alternating group:A4Cyclic group:Z3
A3 in A5Alternating group:A5Cyclic group:Z3
A3 in S4Symmetric group:S4Cyclic group:Z3
A4 in A5Alternating group:A5Alternating group:A4
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2
Cyclic maximal subgroups of quaternion groupQuaternion groupCyclic group:Z4Cyclic group:Z2
D8 in A6Alternating group:A6Dihedral group:D8
D8 in S4Symmetric group:S4Dihedral group:D8
Klein four-subgroup of alternating group:A5Alternating group:A5Klein four-group
Klein four-subgroups of dihedral group:D8Dihedral group:D8Klein four-groupCyclic group:Z2
Non-characteristic order two subgroups of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Cyclic group:Z4
Non-normal Klein four-subgroups of symmetric group:S4Symmetric group:S4Klein four-group
Non-normal subgroups of dihedral group:D8Dihedral group:D8Cyclic group:Z2
S2 in S4Symmetric group:S4Cyclic group:Z2
Subgroup generated by double transposition in symmetric group:S4Symmetric group:S4Cyclic group:Z2
Twisted S3 in A5Alternating group:A5Symmetric group:S3
Z4 in direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z4Cyclic group:Z2

Here are some examples of subgroups in even more complicated/less basic groups not satisfying the property:

 Group partSubgroup partQuotient part
2-Sylow subgroup of general linear group:GL(2,3)General linear group:GL(2,3)Semidihedral group:SD16
Center of central product of D8 and Z4Central product of D8 and Z4Cyclic group:Z4Klein four-group
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-groupKlein four-group
Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Quaternion group
Cyclic maximal subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z8Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z8Cyclic group:Z2
D8 in D16Dihedral group:D16Dihedral group:D8Cyclic group:Z2
Non-central Z4 in M16M16Cyclic group:Z4Cyclic group:Z4
Non-normal subgroups of M16M16Cyclic group:Z2
Q8 in SD16Semidihedral group:SD16Quaternion groupCyclic group:Z2
Q8 in central product of D8 and Z4Central product of D8 and Z4Quaternion groupCyclic group:Z2

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 G^n of G, the corresponding direct power H^n 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.
direct power-closed subgroup property No full invariance is not direct power-closed It is possible to have a fully invariant subgroup H inside a group G and an infinite cardinal \alpha such that the direct power H^\alpha is not a fully invariant subgroup inside the direct power G^\alpha.

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 (see also list of examples) Existentially bound-word subgroup, Image-closed fully invariant subgroup, Intersection of finitely many verbal subgroups, Pseudoverbal subgroup, Quasiverbal subgroup, Quotient-subisomorph-containing subgroup, Weakly image-closed fully invariant subgroup|FULL LIST, MORE INFO
intersection of finitely many verbal subgroups intersection of a finite number of verbal subgroups |FULL LIST, MORE INFO
pseudoverbal subgroup defined as the intersection of normal subgroups for which the quotient group is in a particular pseudovariety Quotient-subisomorph-containing subgroup|FULL LIST, MORE INFO
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 |FULL LIST, MORE INFO
homomorph-containing subgroup contains every homomorphic image homomorph-containing implies fully invariant fully invariant not implies homomorph-containing Intermediately fully invariant subgroup, Sub-homomorph-containing subgroup|FULL LIST, MORE INFO
subhomomorph-containing subgroup contains every homomorphic image of every subgroup (via homomorph-containing) (via homomorph-containing) Homomorph-containing subgroup, Intermediately fully invariant subgroup, Sub-homomorph-containing subgroup, Transfer-closed fully invariant subgroup|FULL LIST, MORE INFO
order-containing subgroup contains every subgroup whose order divides its order (via homomorph-containing) (via homomorph-containing) Homomorph-containing subgroup, Image-closed fully invariant subgroup, Subhomomorph-containing subgroup|FULL LIST, MORE INFO
variety-containing subgroup contains every subgroup in the variety of groups generated by it (via homomorph-containing subgroup) (via homomorph-containing subgroup) Homomorph-containing subgroup, Intermediately fully invariant subgroup, Subhomomorph-containing subgroup, Transfer-closed fully invariant subgroup|FULL LIST, MORE INFO
normal subgroup having no nontrivial homomorphism to its quotient group No nontrivial homomorphism to quotient group Homomorph-containing subgroup, Intermediately fully invariant subgroup, Quotient-subisomorph-containing subgroup|FULL LIST, MORE INFO
normal Hall subgroup normal and Hall: its order and index are relatively prime Complemented fully invariant subgroup, Complemented homomorph-containing subgroup, Homomorph-containing subgroup, Image-closed fully invariant subgroup, Intermediately fully invariant subgroup, Normal subgroup having no nontrivial homomorphism to its quotient group, Order-containing subgroup, Quotient-subisomorph-containing subgroup, Sub-homomorph-containing subgroup, Variety-containing subgroup|FULL LIST, MORE INFO
normal Sylow subgroup normal and Sylow Complemented fully invariant subgroup, Complemented homomorph-containing subgroup, Homomorph-containing subgroup, Image-closed fully invariant subgroup, Intermediately fully invariant subgroup, Normal Hall subgroup, Normal subgroup having no nontrivial homomorphism to its quotient group, Order-containing subgroup, Quotient-subisomorph-containing subgroup, Sub-homomorph-containing subgroup, Variety-containing subgroup|FULL LIST, MORE INFO
quotient-subisomorph-containing subgroup Quotient-subisomorph-containing implies fully invariant Fully invariant not implies quotient-subisomorph-containing Weakly image-closed fully invariant subgroup|FULL LIST, MORE INFO
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 Weakly image-closed fully invariant subgroup|FULL LIST, MORE INFO
intermediately fully invariant subgroup Fully invariant in every intermediate subgroup full invariance does not satisfy intermediate subgroup condition |FULL LIST, MORE INFO
transfer-closed fully invariant subgroup Its intersection with any subgroup is fully invariant in that full invariance does not satisfy transfer condition |FULL LIST, MORE INFO

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) Finite direct power-closed characteristic subgroup, Injective endomorphism-invariant subgroup, Normality-preserving endomorphism-invariant subgroup, Retraction-invariant characteristic subgroup, Strictly characteristic subgroup|FULL LIST, MORE INFO characteristic versus fully invariant
normal subgroup invariant under all inner automorphisms (via characteristic) (via characteristic) Characteristic subgroup, Finite direct power-closed characteristic subgroup, Fully invariant-potentially fully invariant subgroup, Image-potentially fully invariant subgroup, Injective endomorphism-invariant subgroup, Normal-potentially fully invariant subgroup, Normality-preserving endomorphism-invariant subgroup, Potentially fully invariant subgroup, Retraction-invariant characteristic subgroup, Retraction-invariant normal subgroup, Strictly characteristic subgroup|FULL LIST, MORE INFO
strictly characteristic subgroup invariant under all surjective endomorphisms fully invariant implies strictly characteristic strictly characteristic not implies fully invariant Normality-preserving endomorphism-invariant subgroup|FULL LIST, MORE INFO --
injective endomorphism-invariant subgroup invariant under all injective endomorphisms injective endomorphism-invariant not implies fully invariant |FULL LIST, MORE INFO
retraction-invariant subgroup invariant under all retractions Retraction-invariant normal subgroup|FULL LIST, MORE INFO
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 |FULL LIST, MORE INFO
potentially fully invariant subgroup the subgroup is fully invariant in some bigger group Fully invariant-potentially fully invariant subgroup, Normal-potentially fully invariant subgroup|FULL LIST, MORE INFO
finite direct power-closed characteristic subgroup any finite direct 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 Normality-preserving endomorphism-invariant subgroup|FULL LIST, MORE INFO

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 and related observations
potentially operator fully invariant in some larger group potentially fully invariant subgroup by definition; any potentially fully invariant subgroup is normal, but normal not implies potentially fully invariant
intermediately operator fully invariant in every intermediate subgroup intermediately fully invariant subgroup any homomorph-containing subgroup satisfies this property.
image condition operator image is fully invariant in any quotient group image-closed fully invariant subgroup any verbal subgroup satisfies this property.

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 \sigma 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

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 | 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.

State of discourse

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 invariant and fully characteristic are now in vogue.

Resolution of questions that are easy to formulate

Any typical question about the behavior of fully invariant 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 endomorphism monoid that are still open. The reason is that even though not enough is known about the endomorphism monoids, there are other ways to obtain information about the structure of fully invariant subgroups.

At the one extreme, there are abelian groups, where the fully invariant 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 fully invariant subgroups, but the exact nature of the restrictions is hard to work out.

References

Journal references

More info

Textbook references