Class equation decomposition

Proposition (Decomposition underlying the class equation). Let $G$ be a finite group, acting on itself by conjugation. Then:

1. $G$ is a disjoint union of its conjugacy classes.
2. The elements with singleton conjugacy class are exactly the center $Z(G)$.
3. For each $x\in G$, the conjugacy class of $x$ has size $[G:C_G(x)]$, where $C_G(x)$ is the centralizer.

Thus, choosing one representative $x_i$ from each conjugacy class outside the center gives the class equation