# Central 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 Central subgroup, all facts related to Central subgroup) |Survey articles about this | Survey articles about definitions built on this

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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

## Definition

### Symbol-free definition

A subgroup of a group is termed a **central subgroup** if it satisfies the following equivalent conditions:

- 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

A subgroup of a group is termed **central** if it satisfies the following equivalent conditions:

- 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.

## Formalisms

### 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

### Stronger 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 |

### Weaker properties

## Metaproperties

### 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.

### Left-hereditariness

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.

### 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 central subgroups is central. This follows as an easy corollary of the fact that *any* subgroup of a central subgroup is central.

### Join-closedness

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-closedABOUT 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.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

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.

### 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 a central subgroup under a surjective homomorphism is central in the image.

### Trimness

The property of being a central subgroup is trivially true, but is not identity-true. In fact, a group is a central subgroup of itself if and only if it is Abelian.
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