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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.
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RANDOM SUBGROUP PROPERTY: Automorph-conjugate subgroup: A subgroup of a group that is conjugate to any automorphic subgroup. Any characteristic subgroup is automorph-conjugate.

This subgroup property is always true for a subgroup of an Abelian group
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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
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Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Equivalence of definitions= 3 Formalisms 3.1 Function restriction expression 4 Relation with other properties 4.1 Weaker properties 5 Metaproperties 5.1 Left-hereditariness 5.2 Intersection-closedness 5.3 Join-closedness 5.4 Intermediate subgroup condition 5.5 Image condition 5.6 Trimness

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. 
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 can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
View other properties expressible in this formalism OR 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

Weaker properties

• Normal subgroup
• Central factor
• Abelian normal subgroup

Metaproperties

Left-hereditariness

This subgroup property is left-hereditary: any subgroup of a subgroup with this property also has this property

Any subgroup of a central subgroup is central. Thus, the property of being a central subgroup is left-hereditary.

Intersection-closedness

This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property
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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

This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property
View a complete list of join-closed subgroup properties

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

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
View all subgroup properties satisfying the intermediate subgroup condition|View facts related to the 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

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 a complete list of subgroup properties satisfying the 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.