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.