Central subgroup

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
A function restriction expression for the property of being a central subgroup is as follows:

Inner automorphism $$\to$$ Identity map

Metaproperties
Any subgroup of an abelian group is central.

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

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.

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.

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.

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