The centralizer of a subset $S$ of a group$G$ is the set $C_G(S)$ of all elements $c\in G$ such that $c s=s c$ for all $s\in S$. It is the largest subgroup$H$ of $G$ containing $S$ such that $S$ is in the center of $H$. The centralizer of a subset is clearly a subgroup of its normalizer, as fixing the set $g H=H g$ is a weaker requirement than $g h=h g$ for all $h\in H$.