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Bireflectionality in Classical Groups

Published online by Cambridge University Press:  20 November 2018

Erich W. Ellers
Erich W. Ellers*
Affiliation:
University of Toronto, Toronto, Ontario
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The motion groups of the real Euclidean plane and of the elliptic plane, the group of projectivities of a line, the projective general linear group PGL2(K), some orthogonal groups O3(K, Q) with char K = 2 (see [8]), are all bireflectional (zweispiegelig). There can be no doubt that bireflectional groups are of prime importance in any theory of groups that are generated by involutions. A brief look into F. Bachmann's book [1] gives convincing evidence.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 6 , 01 December 1977 , pp. 1157 - 1162
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Bachmann, F., Aufbau der Géométrie aus dem Spiegelungsbegriff, 2nd edition (Springer, New York-Heidelberg-Berlin, 1973).Google Scholar
2. Coxeter, H. S. M., Regular polytopes, 3rd edition (Dover, New York, 1973).Google Scholar
3. Coxeter, H. S. M. Regular complex polytopes (Cambridge University Press, 1974).Google Scholar
4. Dienst, K. J., Bewegungsgruppen projektiv-metrischer Ebenen von Char. 2. J.reine u.angew. Math. 050 (1972), 135–140.Google Scholar
5. Dieudonné, J., La géométrie des groupes classiques (Springer, Berlin-Gôttingen-Heidelberg, 1955).Google Scholar
6. Ellers, E. W., Decomposition of orthogonal, symplectic, and unitary isometries into simple isometries, Abh. Math. Sem. Univ. Hamburg Jfi Google Scholar
7. Ellers, E. W. Decomposition of equiaffinities into reflections, to appear, Geometriae Dedicata.Google Scholar
8. Lingenberg, R.. Die orthogonalen Gruppen Oz(K, Q) ùber Kôrpern von Charakteristik 2, Math. Nachr. 21 (1960), 371–380.Google Scholar
9. O'Meara, O. T.. Group-theoretic characterization of transvections using CDC, Math. Z. 110 (1969), 385–394.Google Scholar
10. Veblen, O. and Young, J. W., Projective geometry, Vol. II (Blaisdell, New York, 1946).Google Scholar
11. Wiener, H., Ùber Gruppen vertauschbarer zweispiegeliger Verwandtschaften, Ber. Verh. kgl. Sachs. Ges. Wiss. Leipzig, math.-nat. Kl. 45 (1893), 555–598.Google Scholar