Let \(G\) be a group and \(h\in G\). The centralizer of \(h\) is \[ Z(h) \coloneqq C_G(h) \coloneqq \theset{ g\in G \st ghg^{-1} = h }. \] For a subset \(H\subseteq G\), the centralizer of \(H\) is \[ Z(H) \coloneqq C_G(H) \coloneqq \theset{g\in G \suchthat ghg^{-1} = h \text{ for all } h\in H} = \Intersect_{h\in H} C_G(h). \]
For the action of \(G\) on itself by conjugation, \(C_G(h)\) is the stabilizer of \(h\), and \(C_G(H)\) is the subgroup of elements that fix every element of \(H\).