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Group-Theoretic Axioms For Projective Geometry

Published online by Cambridge University Press:  20 November 2018

Alex D. Gottlieb  and
Joseph Lipman
Alex D. Gottlieb
Affiliation:
Department of Mathematics Purdue UniversityW. Lafayette IN 47907, U. S. A
Joseph Lipman
Affiliation:
Department of Mathematics Purdue UniversityW. Lafayette IN 47907, U. S. A
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Abstract

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We show that a certain category 𝓖 whose objects are pairs G ⊃ H of groups subject to simple axioms is equivalent to the category of ≥ 2-dimensional vector spaces and injective semi-linear maps; and deduce via the "Fundamental Theorem of Projective Geometry" that the category of ≥ 2-dimensional projective spaces is equivalent to the quotient of a suitable subcategory of 𝓖 by the least equivalence relation which identifies conjugation by any element of H with the identity automorphism of G.

Keywords

groupssemi-linear mapsprojective geometryequivalence of categories15A9920F2951A05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 1 , 01 February 1991 , pp. 89 - 107
Copyright
Copyright © Canadian Mathematical Society 1991

References

[A] Artin, E. Geometric Algebra. Interscience, New York, 1957.Google Scholar
[L] Lipman, J. Definition of affine geometry by a group of transformations, Canad. Math. Bull. 4(1961) 265- 278.Google Scholar