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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $g \\in G$.\nThe inner automorphism given by $g$ is the map $c_g \\colon G \\to G$, $c_g(x) = gxg^{-1}$.\nThe set of all inner automorphisms of $G$ is written $\\operatorname{Inn}(G)$.\n\n## Remarks\n\nEach $c_g$ is an [[automorphism|0K1SCPBNKJ]] of $G$, with inverse $c_{g^{-1}}$.\nSince $c_g \\circ c_h = c_{gh}$, the map $g \\mapsto c_g$ is a homomorphism from $G$ to $\\operatorname{Aut}(G)$, and $\\operatorname{Inn}(G)$ is its image.\nThe subgroup $\\operatorname{Inn}(G)$ is normal in $\\operatorname{Aut}(G)$, because $\\alpha \\circ c_g \\circ \\alpha^{-1} = c_{\\alpha(g)}$ for each automorphism $\\alpha$.\nIn an [[abelian group|082GHNPJ53]], each inner automorphism is the identity map.\n\n## Explanations\n\n### L1 human\n\nEach element of a group gives a symmetry of the group: conjugate everything by this element.\nThese symmetries are called inner, because they come from inside the group.\nIn a group where all elements commute, conjugation changes nothing.\n",
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