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  "body": "\n## Summary\n\nA homogeneous space for a group $G$ is a space on which $G$ acts [[transitively|02GK6HZYBT]], so that all points look alike.\nIn algebraic geometry, it is a variety with a transitive [[action of an algebraic group|RW85C34GQW]].\nSuch a variety is a quotient $G / H$ by the [[stabilizer|0AYMP41THN]] $H$ of a point, at least in characteristic $0$.\nProjective spaces, Grassmannians and [[flag varieties|NHRWCQQKTH]] are homogeneous spaces.\n\n## Explanations\n\n### L1 human\n\nOn a sphere, any point can be moved to any other by a rotation, so the sphere looks the same everywhere.\nA homogeneous space is a space with this property for some group of symmetries.\nIn projective space every point is like every other, because invertible matrices move any line through the origin to any other.\nOnce one point and its stabilizer are known, the whole space is described as the group modulo the stabilizer.\n",
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