Centralizer

Let $G$ be a group and let $S\subseteq G$ be a subset . The centralizer of $S$ in $G$ is $C_G(S) \;=\; \{\,g\in G : gs=sg \text{ for all } s\in S\,\}.$ This is a subgroup of $G$. For a single element $x\in G$, one writes $C_G(x)$ for $C_G(\{x\})$.

Centralizers organize commutation in a group and control conjugacy classes (e.g. via orbit–stabilizer for the conjugation action). The center satisfies $Z(G)=C_G(G)$.

Examples:

• If $G$ is abelian, then $C_G(S)=G$ for every subset $S$.
• In $S_3$, $C_{S_3}((12))=\{e,(12)\}$.
• In $S_3$, $C_{S_3}((123))=\{e,(123),(132)\}$.