# 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 definitionVIEW: Definitions built on this | Facts about this: (factscloselyrelated to Abelian normal subgroup, all facts related to Abelian normal subgroup) |Survey articles about this | Survey articles about definitions built on this

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View a complete list of semi-basic definitions on this wiki

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 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

View a complete list of such conjunctions

## 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 groupVIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

### 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:

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:

## Relation with other properties

### Stronger properties

### Weaker properties

### 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 isnottransitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole groupABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitiveABOUT 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 conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If is an abelian normal subgroup of , is also an abelian normal subgroup in any intermediate subgroup . 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-closedABOUT 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`