Abelian central factor equals central subgroup

Statement
The following are equivalent for a subgroup $$H$$ of a group $$G$$:


 * 1) $$H$$ is a central subgroup of $$G$$: in other words, $$H \le Z(G)$$, or every element of $$H$$ commutes with every element of $$G$$.
 * 2) $$H$$ is an Abelian group, and is a central factor of $$G$$.

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


 * 1) $$HC_G(H) = G$$.
 * 2) Every inner automorphism of $$G$$ restricts to an inner automorphism of $$H$$. The function restriction expression for this is:

Inner automorphism $$\to$$ Inner automorphism

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


 * 1) $$C_G(H) = G$$.
 * 2) Every inner automorphism of $$G$$ restricts to the identity map on $$H$$. The function restriction expression for this is:

Inner automorphism $$\to$$ Identity map