\begin{definition}[Definition:Centralizer/Group Subset]
Let $\struct {G, \circ}$ be a group.
Let $S \subseteq G$.
The '''centralizer of $S$ (in $G$)''' is the set of elements of $G$ which commute with all $s \in S$:
:$\map {C_G} S = \set {x \in G: \forall s \in S: x \circ s = s \circ x}$
\end{definition}