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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]] with [[centre|0DZ93493VN]] $Z(G)$.\nLet $a_1, \\dots, a_k$ be one element from each [[conjugacy class|0DEBXAJ5SA]] of $G$ that has more than one element.\nThen $|G| = |Z(G)| + \\sum_{i=1}^{k} [G : C_G(a_i)]$.\n\n## Remarks\n\nThe conjugacy classes partition $G$, and the classes with one element are exactly the elements of $Z(G)$.\nEach term $[G : C_G(a_i)]$ is the [[size of a class|0VK5V9P8G3]], so it divides $|G|$ and is greater than $1$.\nFor $S_3$, the equation is $6 = 1 + 3 + 2$.\nThe class equation is the key step in the proof that a nontrivial finite $p$-group has a nontrivial centre.\n\n## Explanations\n\n### L1 human\n\nCount the elements of the group class by class.\nThe central elements form classes of size one, and the other classes have sizes that divide the order of the group.\nThis simple count gives strong facts, for example about groups of prime power order.\n",
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