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  "body": "\n## Statement\n\nLet $p$ be a prime number and let $G$ be a finite [[p-group|06QYRTX9KQ]] with $G \\neq \\{e\\}$.\nThen the [[centre|0DZ93493VN]] $Z(G)$ is not $\\{e\\}$.\n\n## Remarks\n\nIn the [[class equation|08BPSC6YTE]] $|G| = |Z(G)| + \\sum [G : C_G(a_i)]$, the left side and each term of the sum are divisible by $p$.\nSo $p$ divides $|Z(G)|$, and $|Z(G)| \\geq p$.\nEquivalently, apply the [[fixed point lemma|098D2E7DEH]] to the action of $G$ on itself by conjugation.\nA consequence is that each finite $p$-group is [[nilpotent|0Q7MMQHC1P]].\n\n## Explanations\n\n### L1 human\n\nThe conjugacy classes of a $p$-group that have more than one element all have sizes divisible by $p$.\nThe order of the group is also divisible by $p$.\nSo the classes with one element, which form the centre, cannot be just the identity.\n",
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