Abelian normal subgroup

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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This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property (itself viewed as a subgroup property): abelian group
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Definition

Symbol-free definition

A subgroup of a group is termed an abelian normal subgroup if it is abelian as a group and normal as a subgroup.

Examples

VIEW: subgroups satisfying this property | subgroups dissatisfying property normal subgroup | subgroups dissatisfying property abelian group
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Generic examples

• The trivial subgroup is an abelian normal subgroup. Further information: trivial subgroup is normal
• The center, and more generally any central subgroup (i.e., any subgroup contained inside the center) is an abelian normal subgroup. Further information: central implies normal
• For a nilpotent group, any member of the second half of the lower central series is an abelian normal subgroup. Further information: Second half of lower central series of nilpotent group comprises abelian groups

Examples in small finite groups

Here are some examples of subgroups in basic/important groups satisfying the property:

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:

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian direct factor	abelian subgroup that is a direct factor	direct factor implies normal		|FULL LIST, MORE INFO
central subgroup	contained in the center	central implies abelian normal	abelian normal not implies central	|FULL LIST, MORE INFO
cyclic normal subgroup	cyclic and a normal subgroup	cyclic implies abelian	abelian not implies cyclic	|FULL LIST, MORE INFO
abelian characteristic subgroup	abelian and a characteristic subgroup	characteristic implies normal	normal not implies characteristic in any nontrivial subquasivariety of the quasivariety of groups	|FULL LIST, MORE INFO
elementary abelian normal subgroup				|FULL LIST, MORE INFO
maximal among abelian normal subgroups	an abelian normal subgroup not contained in any bigger abelian normal subgroup			|FULL LIST, MORE INFO
maximal among abelian characteristic subgroups	an abelian characteristic subgroup not contained in any bigger abelian characteristic subgroup			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
commutator-in-center subgroup	its commutator with the whole group is contained in its center			|FULL LIST, MORE INFO
Dedekind normal subgroup	normal subgroup and also a Dedekind group -- every subgroup is normal in it			|FULL LIST, MORE INFO
class two normal subgroup	normal subgroup that is also a group of nilpotency class two			|FULL LIST, MORE INFO
nilpotent normal subgroup	normal subgroup that is also a nilpotent group			|FULL LIST, MORE INFO
solvable normal subgroup	normal subgroup that is also a solvable group			|FULL LIST, MORE INFO
normal subgroup				|FULL LIST, MORE INFO

Related group properties

The group property of not having any nontrivial abelian normal subgroup is equivalent to the property of being Fitting-free i.e. not having any nontrivial nilpotent normal subgroup.

Facts

• Quotient group acts on abelian normal subgroup: One of the main differences between Abelian normal subgroups and other normal subgroups is that for an Abelian normal subgroup, there is a well-defined action of the quotient group on the subgroup. This is the beginning of group cohomology, which essentially looks at the study of groups that have a given Abelian normal subgroup and a given quotient group, with a specified action of the quotient group on the subgroup.
• Degree of irreducible representation divides index of abelian normal subgroup: For a finite group, the degrees of irreducible representations over an algebraically closed field of characteristic zero divide the index of any Abelian normal subgroup.
• Maximal among abelian normal implies self-centralizing in nilpotent and maximal among abelian normal implies self-centralizing in supersolvable: In a group that is nilpotent or supersolvable, any subgroup that is maximal among Abelian normal subgroups contains its own centralizer.

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

An abelian normal subgroup of an abelian normal subgroup need not be an abelian normal subgroup.

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 an abelian normal subgroup of $G$, $H$ is also an abelian normal subgroup in any intermediate subgroup $K$. This follows from the fact that normality satisfies the same condition: Normality satisfies intermediate subgroup condition.

Image condition

YES: This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property
View other subgroup properties satisfying image condition

The image of an sbelian normal subgroup under a surjective homomorphism is an Abelian normal subgroup of the image. This follows from two facts: Abelianness is quotient-closed, and normality satisfies image condition.

Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closed
ABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness

An intersection of a nonempty collection of Abelian normal subgroups is again an Abelian normal subgroup. This follows from two facts: Abelianness is subgroup-closed, and normality is strongly intersection-closed.

Join-closedness

This subgroup property is not join-closed, viz., it is not true that a join of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not join-closed

A join of Abelian normal subgroups of a group need not be an Abelian normal subgroup. A join of finitely many Abelian normal subgroups, however, is guaranteed to be nilpotent. For full proof, refer: Abelian normal is not join-closed