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: central factor with a group property (itself viewed as a subgroup property): Abelian group
View a complete list of such conjunctions
- It is a subgroup inside the center
- Every inner automorphism of the whole group restricts to the identity map on the subgroup
- Every element of the subgroup commutes with every element of the group
- It is a central factor of the whole group, and is Abelian as a subgroup.
Definition with symbols
- For any in , the map sending in to is the identity map.
- For any in and in , .
- is an Abelian group, and .
Equivalence of definitions
The equivalence of definitions (1), (2) and (3) follows directly from the definition of center. For the equivalence with (4), refer Abelian central factor equals central subgroup.
Function restriction expression
This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property
A function restriction expression for the property of being a central subgroup is as follows:
Inner automorphism Identity map
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|subgroup of abelian group||the whole group is abelian|||FULL LIST, MORE INFO|
|abelian direct factor||the subgroup is abelian and is a direct factor of the whole group|||FULL LIST, MORE INFO|
Tautology when whole group is abelian
This subgroup property is an abelian-tautological subgroup property: it is always true for a subgroup of an abelian group.
View a complete list of abelian-tautological subgroup properties
Any subgroup of an abelian group is central.
This subgroup property is left-hereditary: any subgroup of a subgroup with this property also has this property. Hence, it is also a transitive subgroup property.
Any subgroup of a central subgroup is central. Thus, the property of being a central subgroup is left-hereditary.
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
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An intersection of central subgroups is central. This follows as an easy corollary of the fact that any subgroup of a central subgroup is central.
YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness
The subgroup generated by a family of central subgroups is central. In fact, the subgroup generated by all central subgroups is precisely the center, and this is the largest central 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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ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition
Any central subgroup of a group is also a central subgroup in any intermediate subgroup. Thus, the property of being a central subgroup satisfies the 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
View other subgroup properties satisfying image condition
The image of a central subgroup under a surjective homomorphism is central in the image.