Fully invariant subgroup

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 $H$.
3	(definition in terms of Hom-set maps)	(too complicated to state without symbols)	the image in $\operatorname{Hom}(H,G)$ of the natural map $\operatorname{End}(G) \to \operatorname{Hom}(H,G)$ (by function restriction) is contained in the image in $\operatorname{Hom}(H,G)$ of the natural map $\operatorname{End}(H) \to \operatorname{Hom}(H,G)$ (by inclusion).

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]

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VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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 part	Subgroup part	Quotient part
A3 in S3	Symmetric group:S3	Cyclic group:Z3	Cyclic group:Z2

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

	Group part	Subgroup part	Quotient part
A4 in S4	Symmetric group:S4	Alternating group:A4	Cyclic group:Z2
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group
Center of quaternion group	Quaternion group	Cyclic group:Z2	Klein four-group
Center of special linear group:SL(2,3)	Special linear group:SL(2,3)	Cyclic group:Z2	Alternating group:A4
Center of special linear group:SL(2,5)	Special linear group:SL(2,5)	Cyclic group:Z2	Alternating group:A5
First agemo subgroup of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Klein four-group
First omega subgroup of direct product of Z4 and Z2	Direct product of Z4 and Z2	Klein four-group	Cyclic group:Z2
Klein four-subgroup of alternating group:A4	Alternating group:A4	Klein four-group	Cyclic group:Z3
Normal Klein four-subgroup of symmetric group:S4	Symmetric group:S4	Klein four-group	Symmetric group:S3

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

	Group part	Subgroup part	Quotient part
Center of M16	M16	Cyclic group:Z4	Klein four-group
Center of dihedral group:D16	Dihedral group:D16	Cyclic group:Z2	Dihedral group:D8
Center of nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Klein four-group	Klein four-group
Center of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z2	Dihedral group:D8
D8 in SD16	Semidihedral group:SD16	Dihedral group:D8	Cyclic group:Z2
Derived subgroup of M16	M16	Cyclic group:Z2	Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16	Dihedral group:D16	Cyclic group:Z4	Klein four-group
Derived subgroup of nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Direct product of Z4 and Z2
Direct product of Z4 and Z2 in M16	M16	Direct product of Z4 and Z2	Cyclic group:Z2
Klein four-subgroup of M16	M16	Klein four-group	Cyclic 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 part	Subgroup part	Quotient part
S2 in S3	Symmetric group:S3	Cyclic group:Z2
Z2 in V4	Klein four-group	Cyclic group:Z2	Cyclic group:Z2

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

	Group part	Subgroup part	Quotient part
A3 in A4	Alternating group:A4	Cyclic group:Z3
A3 in A5	Alternating group:A5	Cyclic group:Z3
A3 in S4	Symmetric group:S4	Cyclic group:Z3
A4 in A5	Alternating group:A5	Alternating group:A4
Cyclic maximal subgroup of dihedral group:D8	Dihedral group:D8	Cyclic group:Z4	Cyclic group:Z2
Cyclic maximal subgroups of quaternion group	Quaternion group	Cyclic group:Z4	Cyclic group:Z2
D8 in A6	Alternating group:A6	Dihedral group:D8
D8 in S4	Symmetric group:S4	Dihedral group:D8
Klein four-subgroup of alternating group:A5	Alternating group:A5	Klein four-group
Klein four-subgroups of dihedral group:D8	Dihedral group:D8	Klein four-group	Cyclic group:Z2
Non-characteristic order two subgroups of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Cyclic group:Z4
Non-normal Klein four-subgroups of symmetric group:S4	Symmetric group:S4	Klein four-group
Non-normal subgroups of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2
S2 in S4	Symmetric group:S4	Cyclic group:Z2
Subgroup generated by double transposition in symmetric group:S4	Symmetric group:S4	Cyclic group:Z2
Twisted S3 in A5	Alternating group:A5	Symmetric group:S3
Z4 in direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z4	Cyclic group:Z2

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

	Group part	Subgroup part	Quotient 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 Z4	Central product of D8 and Z4	Cyclic group:Z4	Klein four-group
Center of direct product of D8 and Z2	Direct product of D8 and Z2	Klein four-group	Klein four-group
Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Quaternion group
Cyclic maximal subgroup of dihedral group:D16	Dihedral group:D16	Cyclic group:Z8	Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z8	Cyclic group:Z2
D8 in D16	Dihedral group:D16	Dihedral group:D8	Cyclic group:Z2
Non-central Z4 in M16	M16	Cyclic group:Z4	Cyclic group:Z4
Non-normal subgroups of M16	M16	Cyclic group:Z2
Q8 in SD16	Semidihedral group:SD16	Quaternion group	Cyclic group:Z2
Q8 in central product of D8 and Z4	Central product of D8 and Z4	Quaternion group	Cyclic 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, Right-transitively homomorph-containing 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, Right-transitively homomorph-containing 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, Right-transitively homomorph-containing 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

• 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 invariant 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 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