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The Translation Planes of Dempwolff
Published online by Cambridge University Press: 20 November 2018
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In [2], Dempwolff constructs three translation planes of order 16 using sharply 2-transitive sets of permutations in S16. That is, if acting on Λ is a sharply 2-transitive set of permutations then an affine plane of order n may be defined as follows: The set of points = {(x, y)|x, y ∊ Λ} and the lines = {(x, y)|y = xg for fixed }, {(x, y)|x = c}, {(x, y)|y = c} for c ∊ Λ.
Let V be a vector space of dimension k over F ≅ GF(pr). A translation plane may be defined by finding a set M of pkr – 1 linear transformations such that xy–l is fixed point free on for all x ≠ y in M.
Notice that if we allow V to act on itself then MV is a sharply 2-transitive set on V if and only if xy–1 is fixed point free on for all x ≠ y in M.
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- Copyright © Canadian Mathematical Society 1981
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