The Class Equation

|G| = |Z(G)| + ∑ [G : C_G(x_i)].

The idea

The class equation. For a finite group $G$, $|G| = |Z(G)| + \sum_{i} [G : C_G(x_i)].$ Here $Z(G)$ is the center of $G$, the set of elements that commute with every element of $G$; $C_G(x) = \{g \in G : gx = xg\}$ is the centralizer of $x$, the set of elements that commute with $x$; and the sum runs over one representative $x_i$ from each conjugacy class of size greater than $1$.

The equation records a partition. Call two elements $x$ and $gxg^{-1}$ conjugate; conjugacy is an equivalence relation, so it splits $G$ into disjoint conjugacy classes, and the class sizes add up to $|G|$. The class of $x$ has exactly $[G : C_G(x)]$ elements, so every class size divides $|G|$. A class has size $1$ exactly when its element commutes with all of $G$, and gathering those one-element classes into a single term gives $|Z(G)|$.

The equation turns questions about $G$ into arithmetic: every term on the right divides $|G|$, so once $|G|$ is known, only a few splittings are possible. This constraint forces, for example, a nontrivial center on every group of prime-power order.

Ways to work on it

• Walkthrough. Conjugacy classes, the class equation, and p-group centers.
• Practice. Conjugacy-class size from a centralizer.
• Hardest. Groups of order p^2 are abelian.

Not sure where to start? Take the ten-question placement test.