The Centralizer $C_H(x)$ where $x \in G$ and $H \leq G$. Let $G$ be a group and $H$ be a subgroup of $G$. Let $x \in G$. Then 
$C_H(x)=H$ if and only if $x \in Z(H)$?
It is obvious that if $x \in Z(H)$ then $C_H(x) = H$.
But I could not prove or provide a counter-example to the other statement.
Remark: $C_H(x) = \{h\in H | \ [h,x]=1\}$

Consider the example of $G=\mathbb{Z}_2\times \mathbb{Z}_2$, $H=\mathbb{Z}_2\times \{0\}$. Then $(0,1)$ satisfies $C_H((0,1))=H$, however $(0,1)\notin Z(H)$ because $(0,1)\notin H$.
