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A Note on the Construction of Projective Planes from Groups
Published online by Cambridge University Press: 20 November 2018
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André (1) gave a construction for translation planes from abelian groups possessing “congruences” of subgroups. Schwerdtfeger (3) constructed the plane over a field F from the group 
of substitutions x → ax + b (a, b ∊ F; a ≠ 0). In this note we describe a construction (inspired by Schwerdtfeger's work), from groups, of planes which are duals of near-field planes.
If a plane is (l, m)-transitive (cf. 2, p. 67) for some pair of distinct lines l, m, then the central collineations ϕ with axis m and centre on I may be identified with the “proper” points (that is, points not on I or m) of the plane once an origin O is chosen (not on l or m):
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- Research Article
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- Copyright © Canadian Mathematical Society 1969
References
1.
André, J., Uber nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. Z.
60 (1954), 156–186.Google Scholar
3.
Schwerdtfeger, H., Projective geometry in the one-dimensional affine group, Can. J. Math.
16 (1964), 683–700.Google Scholar
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