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Abelian normal subgroup
From Groupprops
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
Contents |
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.
Relation with other properties
Stronger properties
- Central subgroup (a subgroup contained inside the center)
- Cyclic normal subgroup
- Abelian characteristic subgroup
- Elementary Abelian normal subgroup
- Maximal among Abelian normal subgroups
- Maximal among Abelian characteristic subgroups
Weaker properties
- Commutator-in-center subgroup
- Class two normal subgroup
- Nilpotent normal subgroup
- Solvable normal subgroup
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
Intermediate subgroup condition
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
View all subgroup properties satisfying the intermediate subgroup condition|View facts related to the 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
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 a complete list of subgroup properties satisfying the image condition
The image of an Abelian 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
This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property
View a complete list of intersection-closed subgroup properties
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 an intersection 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
