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  "body": "\n## Definition\n\nLet $X$ be a set and let $G$ be a [[group|G5S1M9]] with a [[transitive action|0WCY42EV9V]] on $X$.\nIn the sense of the *Erlangen program*, the pair $(X, G)$ is a *geometry*.\nA property of subsets of $X$ belongs to the geometry $(X, G)$ if it is invariant under $G$.\nThis means that, for each subset $S$ and each $g \\in G$, it holds for $S$ exactly when it holds for $g(S)$.\nTwo subsets of $X$ are *equivalent* in the geometry if some $g \\in G$ maps one onto the other.\n\n## Remarks\n\nExample: $X = \\R^2$ with $G$ the affine group gives affine geometry.\nThe property \"the points $a, b, c$ are collinear\" belongs to it, since affine maps send lines onto lines.\nNon-example: the property \"the segment $ab$ has length $1$\" does not belong to affine geometry, since the map $x \\mapsto 2x$ is affine and doubles lengths.\nSince $G$ acts transitively, $X$ is a [[homogeneous space|85TWR6E75P]] $G/H$, where $H$ is the stabilizer of a point.\n",
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