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  "body": "\n## Definition\n\nA *finite geometry* is an [[incidence structure|FGKNAFNHA3]] $(P, L, I)$ with a finite set $P$ of points that satisfies the axioms of a given type of geometry.\nThe usual types are [[projective planes|DANFNN8C4Y]], [[affine planes|EC7XT2ZM5S]], and projective and affine spaces of higher dimension.\n\n## Remarks\n\nExample: the [[projective space|ZT9A6JK8QF]] $PG(2, 2)$ over the field with two elements, the [[Fano plane|QQ8GW1ASZM]].\nIt has $7$ points and $7$ lines.\nAny two points lie on exactly one line, and any two lines meet in exactly one point.\nThe points $[1 : 0 : 0]$, $[0 : 1 : 0]$, $[0 : 0 : 1]$, $[1 : 1 : 1]$ have no three on a line, so it is a finite projective plane.\n\nNon-example: the real projective plane satisfies the axioms of a projective plane, but it has infinitely many points, so it is not a finite geometry.\n\nThe term also names the field of mathematics that studies these structures.\nFinite projective spaces of dimension at least $3$ all come from finite fields, by the [[Veblen–Young theorem|NPNNXWMVCC]] and Wedderburn's little theorem; finite planes need not.\n",
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