Complemented normal subgroup

Definition

Definitions in tabular format

No. Shorthand A subgroup of a group is complemented normal if ... A subgroup $H$ of a group $G$ is complemented normal if ...
1 lattice-complemented normal it is normal and lattice-complemented (viz., it possesses a lattice complement) $H \underline{\triangleleft} G$ and there exists a subgroup $K \le G$ satisfying $H \cap K = 1$ and $\langle H,K \rangle = G$
2 permutably complemented normal it is normal and permutably complemented (viz., it possesses a permutable complement) $H \underline{\triangleleft} G$ and there exists a $K \le G$ satisfying $H \cap K = 1$ and $HK = G$
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 $\sigma$ of $G$ such that $\sigma^2 = \sigma$ and $H$ is the kernel of $\sigma$.
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 $H$ is normal in $G$ and the exact sequence: $1 \to H \to G \to G/H \to 1$ splits, hence making $G$ a semidirect product with normal part $H$.

Equivalence of definitions

For full proof, refer: Equivalence of definitions of complemented normal subgroup

Note that any complement to $H$ in $G$ must be isomorphic to the quotient group $G/H$. 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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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 partSubgroup partQuotient part
A3 in S3Symmetric group:S3Cyclic group:Z3Cyclic group:Z2
Z2 in V4Klein four-groupCyclic group:Z2Cyclic 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
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2
Klein four-subgroup of alternating group:A4Alternating group:A4Klein four-groupCyclic group:Z3
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
Normal Klein four-subgroup of symmetric group:S4Symmetric group:S4Klein four-groupSymmetric 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 satisfying the property:

Group partSubgroup partQuotient part
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

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

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
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
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
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
S3 in S4Symmetric group:S4Symmetric group:S3
Subgroup generated by double transposition in symmetric group:S4Symmetric group:S4Cyclic group:Z2
Twisted S3 in A5Alternating group:A5Symmetric group:S3

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 M16M16Cyclic group:Z4Klein four-group
Derived subgroup of M16M16Cyclic 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
Non-normal subgroups of M16M16Cyclic group:Z2

Metaproperties

Note that the notation $K$ as used here is not to be confused with the $K$ 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 $G$ and subgroups $H,K$ of $G$ such that both $H$ and $K$, viewed separately, are complemented normal subgroups of $G$, but $H \cap K$ is not a complemented normal subgroup of $G$.
transitive subgroup property No complemented normal is not transitive it is possible to have groups $H \le K \le G$ such that $H$ is a complemented normal subgroup of $K$ and $K$ is a complemented normal subgroup of $G$, but $H$ is not a complemented normal subgroup of $G$.
intermediate subgroup condition Yes complemented normal satisfies intermediate subgroup condition If $H \le K \le G$ are groups such $H$ is a complemented normal subgroup in $G$, then $H$ is a complemented normal subgroup in $K$.
quotient-transitive subgroup property | Yes complemented normal is quotient-transitive If $H \le K \le G$ are groups such $H$ is a complemented normal subgroup in $G$ and $K/H$ is a complemented normal subgroup in $G/H$, then $K$ is a complemented normal subgroup in $G$.
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

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
direct factor normal subgroup with normal complement Complemented central factor, Complemented transitively normal subgroup, Right-transitively complemented normal subgroup|FULL LIST, MORE INFO
complemented characteristic subgroup characteristic subgroup with permutable complement |FULL LIST, MORE INFO
normal Hall subgroup Hall subgroup (i.e., order and index of subgroup are relatively prime to each other) that is also normal in the whole group normal Hall implies permutably complemented (part of the Schur-Zassenhaus theorem) Complemented fully invariant subgroup, Complemented homomorph-containing subgroup, Complemented isomorph-containing subgroup|FULL LIST, MORE INFO
normal Sylow subgroup Sylow subgroup that is also normal (the whole group needs to be a finite group) (via normal Hall) Complemented fully invariant subgroup, Complemented homomorph-containing subgroup, Complemented isomorph-containing subgroup, Normal Hall subgroup|FULL LIST, MORE INFO
left-transitively complemented normal subgroup Complemented characteristic subgroup|FULL LIST, MORE INFO
right-transitively complemented normal subgroup any complemented normal subgroup of the subgroup is complemented normal in the whole group. |FULL LIST, MORE INFO
complemented central factor central factor with a permutable complement Complemented transitively normal subgroup|FULL LIST, MORE INFO

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 $H$ is a complemented normal subgroup in $G$, and $M$ is an intermediate subgroup, then $H$ is a complemented normal subgroup in $M$. In fact, the retraction for $M$ is simply the restriction to $M$ of the retraction on $G$. To prove that this retraction actually restricts to a well-defined map on $M$, we need to use the fact that $M$ contains $H$. 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 $H \le K \le G$ are groups such that $H$ is a complemented normal subgroup of $G$ and $K/H$ is a complemented normal subgroup of $G/H$. Then, $K$ is a complemented normal subgroup of $G$. 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.
View the GAP code for testing this subgroup property at: IsComplementedNormalSubgroup
View other GAP-codable subgroup properties | View subgroup properties with in-built commands
GAP-codable subgroup property