Centralizer in Subgroup is Intersection

Theorem
Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Then:


 * $\forall x \in G: \map {C_H} x = \map {C_G} x \cap H$

That is, the centralizer of an element in a subgroup is the intersection of that subgroup with the centralizer of the element in the group.

Proof
It is clear that:
 * $g \in \map {C_H} x \iff g \in \map {C_G} x \land g \in H$

The result follows by definition of set intersection.