Abelian normal subgroup

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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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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
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

Generic examples

Examples in small finite groups

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

 Group partSubgroup partQuotient part
A3 in S3Symmetric group:S3Cyclic group:Z3Cyclic group:Z2
Z2 in V4Klein four-groupCyclic group:Z2Cyclic group:Z2

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

 Group partSubgroup partQuotient part
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
Center of quaternion groupQuaternion groupCyclic group:Z2Klein four-group
Center of special linear group:SL(2,3)Special linear group:SL(2,3)Cyclic group:Z2Alternating group:A4
Center of special linear group:SL(2,5)Special linear group:SL(2,5)Cyclic group:Z2Alternating group:A5
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2
Cyclic maximal subgroups of quaternion groupQuaternion groupCyclic group:Z4Cyclic group:Z2
First agemo subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Klein four-group
First omega subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Klein four-groupCyclic group:Z2
Klein four-subgroup of alternating group:A4Alternating group:A4Klein four-groupCyclic group:Z3
Klein four-subgroups of dihedral group:D8Dihedral group:D8Klein four-groupCyclic group:Z2
Non-characteristic order two subgroups of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Cyclic group:Z4
Normal Klein four-subgroup of symmetric group:S4Symmetric group:S4Klein four-groupSymmetric group:S3
Z4 in direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z4Cyclic group:Z2

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

 Group partSubgroup partQuotient part
Center of M16M16Cyclic group:Z4Klein four-group
Center of central product of D8 and Z4Central product of D8 and Z4Cyclic group:Z4Klein four-group
Center of dihedral group:D16Dihedral group:D16Cyclic group:Z2Dihedral group:D8
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-groupKlein four-group
Center of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Klein four-groupKlein four-group
Center of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z2Dihedral group:D8
Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Quaternion group
Cyclic maximal subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z8Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z8Cyclic group:Z2
Derived subgroup of M16M16Cyclic group:Z2Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z4Klein four-group
Derived subgroup of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Direct product of Z4 and Z2
Direct product of Z4 and Z2 in M16M16Direct product of Z4 and Z2Cyclic group:Z2
Klein four-subgroup of M16M16Klein four-groupCyclic group:Z4
Non-central Z4 in M16M16Cyclic group:Z4Cyclic group:Z4
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Dihedral group:D8

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 Central subgroup|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 Homocyclic normal subgroup|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 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 Normal subgroup whose inner automorphism group is central in automorphism group|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 Commutator-in-center subgroup, Normal subgroup whose inner automorphism group is central in automorphism group|FULL LIST, MORE INFO
nilpotent normal subgroup normal subgroup that is also a nilpotent group Class two normal subgroup, Dedekind normal subgroup|FULL LIST, MORE INFO
solvable normal subgroup normal subgroup that is also a solvable group Dedekind normal subgroup, Nilpotent normal subgroup|FULL LIST, MORE INFO
normal subgroup Class two normal subgroup, Commutator-in-center subgroup, Dedekind normal subgroup, Nilpotent normal subgroup, Normal subgroup whose inner automorphism group is central in automorphism group, Solvable 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

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