The centralizer $C_G(H)$ of a subgroup

Let $G$ be a group (Group and abelian group) and let $H\le G$. The centralizer of $H$ in $G$ is

$C_G(H):=\{g\in G:gh=hg\text{ for every }h\in H\}\mathrm{.}$

Equivalently, $C_G(H)={\bigcap }_{h\in H}{C}_G(h)$, the intersection of the centralizers of the individual elements of $H$ (The conjugacy class ${\mathrm{Cl}}_G(x)$ and centralizer $C_G(x)$ of an element).

Why it is a subgroup. The identity satisfies $eh=h=he$ for every $h\in H$, so $e\in C_G(H)$. If $g_1,g_2\in C_G(H)$ and $h\in H$, then

$(g_1{g}_2)h=g_1(g_{2}h)=g_1(h{g}_2)=(g_{1}h)g_2=(h{g}_1)g_2=h(g_1{g}_2),$

so $g_1{g}_2\in C_G(H)$. If $g\in C_G(H)$ and $h\in H$, then multiplying $gh=hg$ by $g^{-1}$ on both sides gives $h{g}^{-1}=g^{-1}h$, so $g^{-1}\in C_G(H)$. Hence $C_G(H)\le G$.

Two special cases are used without further comment. Taking $H=G$ gives $C_G(G)=Z(G)$, the center (The center $Z(G)$ of a group). Intersecting with $H$ gives $C_G(H)\cap H=Z(H)$, since an element of $H$ lies in $C_G(H)$ exactly when it commutes with every element of $H$.