Complemented normal subgroup
Definition
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]
This is a variation of normal subgroup|Find other variations of normal subgroup | Read a survey article on varying normal subgroup
Examples
Extreme examples
- 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
If the whole group is an abelian group, being a complemented normal subgroup is equivalent to being a direct factor, i.e., a part of an internal direct product.
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. |
General examples
- 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 | |
---|---|---|---|
A3 in S3 | Symmetric group:S3 | Cyclic group:Z3 | Cyclic group:Z2 |
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:
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 |
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
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
Stronger properties
Weaker 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 |
Related properties
Retract is a (not necessarily normal) subgroup that has a permutable complement which is a normal subgroup.
Metaproperties
Intersection-closedness
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
Quotient-transitivity
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
Applying the left transiter to this property gives: left-transitively complemented normal subgroup
The right transiter
Applying the right transiter to this property gives: right-transitively complemented normal subgroup
Testing
GAP code
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