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 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]

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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:

1. It is a subgroup inside the center
2. Every inner automorphism of the whole group restricts to the identity map on the subgroup
3. Every element of the subgroup commutes with every element of the group
4. It is a central factor of the whole group, and is Abelian as a subgroup.

Definition with symbols

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

1. $H \le Z(G)$
2. For any $x$ in $G$, the map sending $h$ in $H$ to $xhx^{-1}$ is the identity map.
3. For any $x$ in $G$ and $h$ in $H$, $xh=hx$.
4. $H$ is an Abelian group, and $HC_G(H) = G$.

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 $\to$ 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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup				Abelian normal subgroup, Amalgam-characteristic subgroup, Amalgam-strictly characteristic subgroup, Center-fixing automorphism-invariant subgroup, Central factor, Class two normal subgroup, Commutator-in-center subgroup, Conjugacy-closed normal subgroup, Dedekind normal subgroup, Direct factor over central subgroup, Hereditarily normal subgroup, Join-transitively central factor, Nilpotent normal subgroup, SCAB-subgroup, Transitively normal subgroup|FULL LIST, MORE INFO
central factor	product with centralizer is whole group			Intersection-transitively central factor, Join of direct factor and central subgroup, Join-transitively central factor|FULL LIST, MORE INFO
hereditarily normal subgroup (also called quasicentral subgroup)	every subgroup of it is normal in the whole group			Abelian hereditarily normal subgroup|FULL LIST, MORE INFO
SCAB-subgroup	every subgroup-conjugating automorphism of the whole group restricts to a subgroup-conjugating automorphism of the subgroup			Central factor, Hereditarily normal subgroup, Join-transitively central factor|FULL LIST, MORE INFO
transitively normal subgroup	every normal subgroup of it is normal in the whole group			Abelian hereditarily normal subgroup, Central factor, Conjugacy-closed normal subgroup, Hereditarily normal subgroup, Join-transitively central factor, SCAB-subgroup|FULL LIST, MORE INFO
conjugacy-closed subgroup	any two elements in it that are conjugate in the whole group are conjugate in the subgroup			Central factor, Conjugacy-closed normal subgroup, Subset-conjugacy-closed subgroup|FULL LIST, MORE INFO
conjugacy-closed normal subgroup	normal and conjugacy-closed			Central factor, Join-transitively central factor|FULL LIST, MORE INFO
abelian normal subgroup	abelian as a group and normal in the whole group			|FULL LIST, MORE INFO

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-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.

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-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.
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

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
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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. Bold text