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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] with [[centre|0DZ93493VN]] $Z(G)$.\nThen the map $gZ(G) \\mapsto c_g$, where $c_g(x) = gxg^{-1}$, is an [[isomorphism|08MG7H8JW5]] from $G/Z(G)$ to $\\operatorname{Inn}(G)$.\n\n## Remarks\n\nThe map $g \\mapsto c_g$ is a surjective homomorphism from $G$ to $\\operatorname{Inn}(G)$. Its kernel is $Z(G)$, because $c_g$ is the identity exactly when $g$ commutes with each element.\nThe [[first isomorphism theorem|0CXXJNG5B6]] then gives the result.\nFor example, the symmetric group $S_3$ has trivial centre, so $\\operatorname{Inn}(S_3) \\cong S_3$.\n\n## Explanations\n\n### L1 human\n\nTwo elements give the same conjugation exactly when they differ by an element of the centre.\nSo the inner automorphisms are counted by the cosets of the centre.\nA group with a big centre has few inner automorphisms.\n",
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  "name": "Inner automorphism group is isomorphic to G modulo its centre",
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