The Class Equation

The Class Equation is a result in group theory. It states that

$|G|=|Z(G)|+\sum_{i=1}^r|G : C_{G}(g_i)|$

where $g_i$ are representatives of the conjugacy classes not in the center of $G$ .

Proof: Let $A\le G$ . We define $C_G(A)=\{g\in G~|~gag^{-1}=a\}$ for all $a\in A$ . The number of conjugates of an element $s\in G$ is the index of the centralizer of $s$ , denoted as $|G : C_G(s)|$ . The element $\{x\}$ is a conjugacy class of order $1$ as long as $x\in Z(G)$ , since then $gxg^{-1}=x$ for all $g\in G$ . Let $Z(G)=\{1,z_1,z_2,\ldots, z_m\}$ and $\mathcal{K}_1,\mathcal{K}_2,\ldots, \mathcal{K}_r$ denote the conjugacy classes not in the center. Let $g_i$ be a representative of each conjugacy class $\mathcal{K}_i$ (which is just an element in the conjugacy class). Since these all partition $G$ , we have $\begin{align*} |G|&=|Z(G)|+\sum_{i=1}^r|\mathcal{K}_i|\\ &=|Z(G)|+\sum_{i=1}^r|G : C_{G}(g_i)| \end{align*}$ which is exactly what we wanted to show $\square$

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The Class Equation