= Class equation
{title2=$|G|=|Z(G)|+\sum_i[G:C_G(x_i)]$}
{wiki}
For a <finite group> $G$, choose one representative $x_i$ from each <conjugacy class> with more than one element. The <class equation> is
$$
|G|=|Z(G)|+\sum_i[G:C_G(x_i)]=|Z(G)|+\sum_i\frac{|G|}{|C_G(x_i)|},
$$
where $Z(G)$ is the <center of a group> and $C_G(x_i)$ is the <centralizer> of $x_i$. Each central element contributes its own one-element <conjugacy class>. Every remaining <conjugacy class> contributes its size, given by the <orbit-stabilizer theorem>. The sum counts each element exactly once. See https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/96221ce2d5b1077b3e5d3c5e08c88afe_MIT18_703S13_pra_l_23.pdf[McKernan, Group actions and automorphisms, lemma 23.6].
For <conjugacy classes of the symmetric group on three letters>, take $x_1=(12)$ and $x_2=(123)$. Their <centralizers> have sizes two and three, and the <center of a group> has size one, giving
$$
6=1+\frac62+\frac63=1+3+2.
$$
For an <abelian group>, every <centralizer> is the whole <group>, every <conjugacy class> has size one, and the sum over noncentral representatives is empty.
The power of this counting identity is that the sizes of <conjugacy classes> constrain <group> structure. For a nontrivial finite <p-group> with <group order> $p^n$, every noncentral <conjugacy class> has size a positive power of $p$, by <Lagrange's theorem>. The <class equation> therefore gives $|Z(G)|\equiv0\pmod p$. Since the <identity element> lies in the <center of a group>, the <center of a group> must have at least $p$ elements. This obtains an element commuting with the entire <group> without constructing one. See https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/96221ce2d5b1077b3e5d3c5e08c88afe_MIT18_703S13_pra_l_23.pdf[McKernan, lemma 23.7].
Historically, these ideas belong to the nineteenth-century development of <permutation groups> and <normal subgroups>. Milne's historical opening records Galois's introduction of <normal subgroups> in 1832 and Jordan's systematic treatment of <permutation groups> in 1870. This is context for the <conjugation action> and the <class equation>, rather than a claim that either author first stated the modern formula.<Group Theory (v4.01) by James S. Milne>{ref}
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