Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-31T14:40:26.211Z Has data issue: false hasContentIssue false
  • English
  • Français

Collineation Groups of Generalized André Planes

Published online by Cambridge University Press:  20 November 2018

David A. Foulser
David A. Foulser*
Affiliation:
University of Illinois at Chicago Circle, Chicago, Illinois
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In a previous paper (5), I constructed a class of translation planes, called generalized André planes or λ-planes, and discussed the associated autotopism collineation groups. The main question unanswered in (5) is whether or not there exists a collineation η of a λ-plane Π which moves the two axes of Π but does not interchange them.

The answer to this question is “no”, except if Π is a Hall plane (or possibly if the order n of Π is 34) (Corollary 2.8). This result makes it possible to determine the isomorphisms between λ-planes. More specifically, let Π and Π′ be two λ-planes of order n coordinatized by λ-systems Qand Q′, respectively. Then, except possibly if n = 34, Π and Π′ are isomorphic if and only if Q and Q′ are isotopic or anti-isotopic (Corollary 2.13). In particular, Π is an André plane if and only if Q is an André system (Corollary 2.14).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 358 - 369
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. André, J., Projective Ebenen über Fastkörpern, Math. Z. 62 (1955), 137–160.Google Scholar
2. Artin, E., Galois theory, 2nd ed. (North State Press, Hammond, Indiana, 1964).Google Scholar
3. Buekenhout, F., Les plans d’ Andre finis de dimension quatre sur le noyau, Acad. Roy. Belg. Bull. CI. Sci. 50 (1964), 446–457.Google Scholar
4. Foulser, D., Solvable flag transitive affine groups, Math. Z. 86 (1964), 191–204.Google Scholar
5. Foulser, D., A generalization of Andre's systems, Math. Z. 100 (1967), 380–395.Google Scholar
6. Foulser, D., A class of translation planes, U(QB), Math. Z. 101 (1967), 95–102.Google Scholar
7. Hughes, D. R., Collineation groups of non-Desarguesian planes. I. Amer. J. Math. 81 (1959), 921–938 Google Scholar
8. Ostrom, T. G., Translation planes and configurations in Desarguesian planes, Arch. Math. 11 (1960), 457–464.Google Scholar
9. Ostrom, T. G., Nets with critical deficiency, Pacific J. Math. 14 (1964), 1381–1387,Google Scholar
10. Ostrom, T. G., Replaceable nets, net collineations, and net extensions, Can. J. Math. 18 (1966), 201–209.Google Scholar
11. Ostrom, T. G., Vector spaces and construction of finite projective planes, Arch. Math. 19 (1968), 1–25.Google Scholar
12. Ostrom, T. G., Partial Desarguesian planes, Math. Z. 106 (1968), 113–122.Google Scholar
13. Panella, G., Le collineazioni nei piani di Marshall Hall, Riv. Mat. Univ. Parma 1 (1960), 171–184.Google Scholar