Central element of a group ($g\in Z(G)⟺gx=xg\text{for all }x\in G$)

An element of a group is central when it commutes with every element of that group. The set of all such elements is the center of a group. Equivalently, its conjugacy class is a singleton: $xg{x}^{-1}=g$ for every $x$ is exactly the commutation condition. This lets centrality identify one-element conjugacy classes without enumerating every conjugation.