Complemented normal subgroup
Definitions in tabular format
|No.||Shorthand||A subgroup of a group is complemented normal if ...||A subgroup of a group is complemented normal if ...|
|1||lattice-complemented normal||it is normal and lattice-complemented (viz., it possesses a lattice complement)||and there exists a subgroup satisfying and|
|2||permutably complemented normal||it is normal and permutably complemented (viz., it possesses a permutable complement)||and there exists a satisfying and|
|3||kernel of retraction||It occurs as the kernel of a retraction (i.e., an endomorphism that equals its own square)||there is an endomorphism of such that and is the kernel of .|
|4||split exact sequence, normal part of semidirect product||it is normal and the exact sequence corresponding to this normal subgroup splits i.e. it is the normal subgroup part in an internal semidirect product||is normal in and the exact sequence: splits, hence making a semidirect product with normal part .|
Equivalence of definitions
For full proof, refer: Equivalence of definitions of complemented normal subgroup
Note that any complement to in must be isomorphic to the quotient group . For full proof, refer: Complement to normal subgroup is isomorphic to quotient
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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- Every group is a complemented normal subgroup of itself, with the complement being the trivial subgroup.
- The trivial subgroup is a complemented normal subgroup in any group, with the complement being the whole group.
Examples in abelian groups
|Type of group||High occurrence or low occurrence?||Complemented normal subgroups||Explanation/comment|
|cyclic group of prime power order||low||whole group and trivial subgroup||it is hard to find complements because the coset of the generator contains only generators.|
|finite cyclic group||relatively low||only the Hall subgroups are complemented normal|
|elementary abelian group, or more generally, additive group of a field||high||all subgroups||every subspace of a vector space has a complement.|
- Low occurrence: A splitting-simple group is a nontrivial group with no proper nontrivial complemented normal subgroup. Any simple group is splitting-simple, but there exist splitting-simple groups that are not simple.
- High occurrence: A C-group is a group in which every subgroup is permutably complemented, and hence, every normal subgroup is a complemented normal subgroup.
Subgroups satisfying the property
Here are some examples of subgroups in basic/important groups satisfying the property:
|Group part||Subgroup part||Quotient part|
|Z2 in V4||Klein four-group||Cyclic group:Z2||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:
|Group part||Subgroup part||Quotient part|
|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|
Subgroups not satisfying the property
Here are some examples of subgroups in basic/important groups not satisfying the property:
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:
Note that the notation as used here is not to be confused with the used to denote the complement in the definition as presented above.
|Metaproperty name||Satisfied?||Proof||Statement with symbols|
|finite-intersection-closed subgroup property||No||same example as used for direct factor is not intersection-closed||It is possible to have a group and subgroups of such that both and , viewed separately, are complemented normal subgroups of , but is not a complemented normal subgroup of .|
|transitive subgroup property||No||complemented normal is not transitive||it is possible to have groups such that is a complemented normal subgroup of and is a complemented normal subgroup of , but is not a complemented normal subgroup of .|
|intermediate subgroup condition||Yes||complemented normal satisfies intermediate subgroup condition||If are groups such is a complemented normal subgroup in , then is a complemented normal subgroup in .|
|quotient-transitive subgroup property | Yes||complemented normal is quotient-transitive||If are groups such is a complemented normal subgroup in and is a complemented normal subgroup in , then is a complemented normal subgroup in .|
|trim subgroup property||Yes||(obvious)||In any group, the whole group and the trivial subgroup are complemented normal subgroups.|
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|endomorphism kernel||kernel of an endomorphism||complemented normal implies endomorphism kernel||endomorphism kernel not implies complemented normal||Template:Intermediate notion short|
|normal subgroup||Endomorphism kernel, Intermediately endomorphism kernel, Normal subgroup having a 1-closed transversal, Powering-invariant normal subgroup, Quotient-powering-invariant subgroup|FULL LIST, MORE INFO|
|permutably complemented subgroup||has a permutable complement|||FULL LIST, MORE INFO|
|lattice-complemented subgroup||has a lattice complement||Permutably complemented subgroup|FULL LIST, MORE INFO|
|normal subgroup having a 1-closed transversal||normal, has a left transversal that is a 1-closed subset|||FULL LIST, MORE INFO|
This subgroup property is not intersection-closed, viz., it is not true that an intersection of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not intersection-closed
The intersection of two complemented normal subgroups need not be a complemented normal subgroup. The proof of this relies on the same example which shows that direct factor is not intersection-closed.
Intermediate subgroup condition
YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition
If is a complemented normal subgroup in , and is an intermediate subgroup, then is a complemented normal subgroup in . In fact, the retraction for is simply the restriction to of the retraction on . To prove that this retraction actually restricts to a well-defined map on , we need to use the fact that contains . For full proof, refer: Complemented normal satisfies intermediate subgroup condition
This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties
Suppose are groups such that is a complemented normal subgroup of and is a complemented normal subgroup of . Then, is a complemented normal subgroup of . For full proof, refer: Complemented normal is quotient-transitive
Effect of property operators
The left transiter
The right transiter
One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.GAP-codable subgroup property
View the GAP code for testing this subgroup property at: IsComplementedNormalSubgroup
View other GAP-codable subgroup properties | View subgroup properties with in-built commands