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 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:

	Group part	Subgroup part	Quotient part
A4 in S4	Symmetric group:S4	Alternating group:A4	Cyclic group:Z2
Cyclic maximal subgroup of dihedral group:D8	Dihedral group:D8	Cyclic group:Z4	Cyclic group:Z2
Klein four-subgroup of alternating group:A4	Alternating group:A4	Klein four-group	Cyclic group:Z3
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
Normal Klein four-subgroup of symmetric group:S4	Symmetric group:S4	Klein four-group	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 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:

	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:

	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
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group
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
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
S3 in S4	Symmetric group:S4	Symmetric group:S3
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

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 M16	M16	Cyclic group:Z4	Klein four-group
Derived subgroup of M16	M16	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
Non-normal subgroups of M16	M16	Cyclic 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
regular kernel				|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