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 of a group is termed a fully invariant subgroup of if ... |
---|---|---|---|
1 | endomorphism-invariant | it is invariant under all endomorphisms of the whole group. | for any endomorphism of , or equivalently, for all . |
2 | endomorphism restricts to endomorphism | every endomorphism of the whole group restricts to an endomorphism of the subgroup. | for any endomorphism of , and the restriction of to is an endomorphism of . |
3 | (definition in terms of Hom-set maps) | (too complicated to state without symbols) | the image in of the natural map (by function restriction) is contained in the image in of the natural map (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]
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
- The trivial subgroup is always fully invariant.
- Every group is fully invariant as a subgroup of itself.
Examples
- 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 powers, for some choice of , and such a set is clearly invariant under endomorphisms. (In fact, it is a verbal subgroup).
- More generally, in any abelian group, the set of powers is a verbal subgroup, and hence fully invariant. The set of elements whose order divides is also fully invariant, though not necessarily verbal (for instance, in the group of all roots of unity, the subgroup of roots for fixed is fully invariant but not verbal).
- 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
- 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).
- 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:
Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:
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:
Here are some examples of subgroups in even more complicated/less basic groups not satisfying the property:
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
transitive subgroup property | Yes | full invariance is transitive | If , with fully invariant in and fully invariant in , then is fully invariant in . |
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 is a fully invariant subgroup inside but is not a fully invariant subgroup of . |
strongly intersection-closed subgroup property | Yes | full invariance is strongly intersection-closed | If , are all fully invariant subgroups of , then is also fully invariant in . |
strongly join-closed subgroup property | Yes | full invariance is strongly join-closed | If , are all fully invariant subgroups of , then is also fully invariant in . |
commutator-closed subgroup property | Yes | full invariance is commutator-closed | If are fully invariant subgroups of , so is . |
quotient-transitive subgroup property | Yes | full invariance is quotient-transitive | If such that is fully invariant in and is fully invariant in , then is fully invariant in . |
finite direct power-closed subgroup property | Yes | full invariance is finite direct power-closed | If is fully invariant in , then in any finite direct power of , the corresponding direct power is fully invariant. |
restricted direct power-closed subgroup property | Yes | full invariance is restricted direct power-closed | If is fully invariant in , then in any restricted direct power of , the corresponding direct power of 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 inside a group and an infinite cardinal such that the direct power is not a fully invariant subgroup inside the direct power . |
Relation with other properties
Stronger properties
Weaker properties
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 of a group is termed fully characteristic if:
The condition in parentheses is a verification that the function is an endomorphism of .
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 | is a fully invariant subgroup of if ... | This means that full invariance is ... | Additional comments |
---|---|---|---|
endomorphism function | every endomorphism of sends every element of to within | the invariance property for endomorphisms | |
endomorphism endomorphism | every endomorphism of restricts to an endomorphism of | the balanced subgroup property for endomorphisms | Hence, it is a t.i. subgroup property, both transitive and identity-true |
endomorphism endomorphism | every endomorphism of restricts to an endomorphism of | 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