Definition:Central Subgroup

Theorem
Let $G$ be a group.

Then every subgroup of $G$ which is a subset of the center of $G$ is a normal subgroup of $G$ and is abelian.

Such a subgroup is called a central subgroup of $G$.

Proof

 * Let $H \le G, H \subseteq Z \left({G}\right)$.

Then:


 * The fact that $H$ is abelian follows from the fact that $Z \left({G}\right)$ is itself abelian.