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Finite Generalized Andre Planes
The Andre Planes
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2-group 2-subgroup of G abelian group acts doubly transitively acts transitively affine plane affine point assume automorphism axis Baer subplane bijection centre collineation group conjugacy classes conjugate contradiction Corollary cyclic subgroup defined denote desarguesian plane distinct points divides q division ring elementary abelian elements of order exists exactly fixed point follows Furthermore G acts group of order Hence there exists implies incidence structure induces infer involution involutory homology isomorphic to SL(2,5 K(oo left invariant Lemma length q Let G Let Q line orbits mapping Maschke's theorem Moreover nearfield plane normal subgroup obtain operates transitively orbit of length order q plane of order point orbit points of 21 projective plane Proof proves quasifield quaternion group Schur's lemma set of points subgroup of order subspaces of rank Sylow 2-subgroup tangent Theorem translation plane vector space whence yields