The Class Equation

Let \(X\) be a finite \(G\)-set and \(X_G\) be the set of fixed points in \(X\); that is, \[\begin{aligned}\end{aligned}\]. Since the orbits of the action partition \(X\), \[\begin{aligned}\end{aligned}\], where \(x_k, \ldots, x_n\) are representatives from the distinct nontrivial orbits of \(X\).

Now consider the special case in which \(G\) acts on itself by conjugation, \((g,x) \mapsto gxg^{-1}\). The center of \(G\), \[\begin{aligned}\end{aligned}\], is the set of points that are fixed by conjugation. The nontrivial orbits of the action are called the conjugacy classes of \(G\). If \(x_1, \ldots, x_k\) are representatives from each of the nontrivial conjugacy classes of \(G\) and \(|{\mathcal O}_{x_1}| = n_1, \ldots, |{\mathcal O}_{x_k}| = n_k\), then \[\begin{aligned}\end{aligned}\]. The stabilizer subgroups of each of the \(x_i\)'s, \(C(x_i) = \{ g \in G: g x_i = x_i g \}\), are called the centralizer subgroups of the \(x_i\)'s. From , we obtain the class equation: \[\begin{aligned}\end{aligned}\]. One of the consequences of the class equation is that the order of each conjugacy class must divide the order of \(G\).

Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.