Abelian central factor equals central subgroup

This article gives a proof/explanation of the equivalence of multiple definitions for the term central subgroup
View a complete list of pages giving proofs of equivalence of definitions

Statement

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

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

Definitions used

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

Proof

Proof in the language of centralizers

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Using function restriction expressions

This subgroup property implication can be proved by using function restriction expressions for the subgroup properties
View other implications proved this way |read a survey article on the topic