centralizer

Let $G$ a group acting on itself by conjugation.Let $X$ be a subset of $G$.The stabilizer of $X$ is called the centralizer of $X$ and it’s the set

$$C_G(X)=\{g\in G:gx{g}^{-1}=x\mathit{\hspace{1em}}\text{for all }x\in X\}$$

For any group $G$, $C_G(G)=Z(G)$, the center of $G$. Thus, any subgroup of $C_G(G)$ is an abelian subgroup of $G$. However, the converse is generally not true. For example, take any non-abelian group and pick any element not in the center. Then the subgroup generated by it is obviously abelian, clearly non-trivial and not contained in the center.