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]
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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VIEW: subgroups satisfying this property | subgroups dissatisfying property normal subgroup | subgroups dissatisfying property abelian group
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- 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
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.
- 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.
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
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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.
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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.
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
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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.
YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
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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.
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