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Projective Geometry in the One-Dimensional Affine Group

Published online by Cambridge University Press:  20 November 2018

Hans Schwerdtfeger
Hans Schwerdtfeger*
Affiliation:
McGill University, Montreal
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The idea of considering the set of the elements of a group as a space, provided with a topology, measure, or metric, connected somehow with the group operation, has been used often in the work of E. Cartan and others. In the present paper we shall study a very special group whose space can be embedded naturally into a projective plane and where the straight lines have a very simple group-theoretical interpretation. It remains to be seen whether this geometrical embedding in a projective space can be extended to other classes of groups and whether the method could become an instrument of geometrical investigation, like co-ordinates or reflections. In the final section it is shown how a geometrical theorem may lead to relations within the group.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 683 - 700
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Menger, K., Frammenti piani autoduali e relative sostituzioni, Atti Accad. Naz. Lincei, Rend. CI. Sci. Mat. Nat. (8), 30 (1961), 713–717. (Cf. Math. Rev., U(1962), No. 4A 2267.)Google Scholar
2. Schwerdtfeger, H., Über eine spezielle Klasse Frobenius’ scher Gruppen, Arch. Math., 13 (1962), 283–289. (Cf. Math. Rev. 26 (1963), No 4, 3770.)Google Scholar
3. Veblen, O. and Young, J. W., Projective Geometry', vol. I (Boston, 1910).Google Scholar