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  "body": "\n## Definition\n\nLet $k$ be an algebraically closed field, and let $G$ be an [[algebraic group|GH81SYAPPN]] over $k$.\nA *homogeneous space* for $G$ is a nonempty variety $X$ over $k$ with an [[action|RW85C34GQW]] of $G$ such that $G(k)$ acts [[transitively|02GK6HZYBT]] on $X(k)$.\n\n## Remarks\n\nLet $x \\in X(k)$ and let $G_x$ be its [[stabilizer|0AYMP41THN]], a closed subgroup of $G$.\nThe orbit map $g \\mapsto g \\cdot x$ induces a bijective morphism $G / G_x \\to X$.\nIn characteristic $0$ this morphism is an isomorphism, so $X \\cong G / G_x$.\nIn characteristic $p$ it can fail to be an isomorphism, because the orbit map need not be separable.\n\nExample: $\\mathrm{GL}_{n+1}$ acts on $\\mathbb{P}^n$ by $g \\cdot [v] = [g v]$.\nAny nonzero vector can be sent to any other by an invertible matrix, so the action is transitive and $\\mathbb{P}^n$ is a homogeneous space.\n\nNon-example: $\\mathbb{G}_m$ acts on $\\mathbb{A}^1$ by $t \\cdot x = t x$.\nThere are two orbits, $\\set{0}$ and $\\mathbb{A}^1 \\setminus \\set{0}$, so the action is not transitive.\n\nIn other settings, such as Lie groups and topological groups, a homogeneous space is a manifold or a space with a transitive action of the group.\n",
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