Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-06T04:12:26.268Z Has data issue: false hasContentIssue false
  • English
  • Français

On Translation Planes with Affine Central Collineations, II

Published online by Cambridge University Press:  20 November 2018

Norman L. Johnson  and
Michael J. Kallaher
Norman L. Johnson
Affiliation:
University of Iowa, Iowa City, Iowa
Michael J. Kallaher
Affiliation:
University of Iowa, Iowa City, Iowa
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.

This article, as the name implies, is a continuation of [9]. In that article the second author investigates finite translation planes containing both affine dations and affine homologies. (See the beginning of Section 2 for definitions.) Such translation planes are called Ei7-planes. In [9] the author restricted himself to translation planes of characteristic p ≧ 5. The main reasons for this were that Ostrom's and Hering's theorem [13;4] on affine dations excluded the case p = 3 and the conclusions were easier to interpret geometrically when p ≧ 5 (as opposed to the case p = 2). Since then Ostrom [17] has settled the case p = 3.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 1 , 01 February 1976 , pp. 116 - 129
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. André, J., Ûber nicht-Desarguessche Ebenen mit transitiver Translationgruppe, Math. Z. 60 (1954) 156–186.Google Scholar
2. Burmeister, M. V. D. and Hughes, D. R., On the solvability of autotopism groups, Arch. Math. 16 (1965), 178–183.Google Scholar
3. Dembowski, P., Finite geometries (Springer-Verlag, Berlin-Heidelberg, 1968).Google Scholar
4. Hering, C., On shears of translation planes, Abh. Math. Sem. Univ., Hamburg, 37 (1972), 258–268.Google Scholar
5. Hering, C., On projective planes of type VI, to appear.Google Scholar
6. Hua, L. K., On the automorphisms of the symplectic group over any field, Ann. of Math. 49 (1948), 739–759.Google Scholar
7. Hughes, D. R. and Piper, F. C., Projective planes (Springer-Verlag, New York-Heidelberg- Berlin, 1973).Google Scholar
8. Johnson, N. L. and Ostrom, T. G., Translation planes with several homology or elation groups of order 3, Geometriae Dedicata 2 (1973), 65–81.Google Scholar
9. Kallaher, M. J., On translation planes with affine central collineations, to appear in Geometriae Dedicata 4 (1975), 71–90.Google Scholar
10. Kallaher, M.J. and Ostrom, T. G., Fixed point free groups, rank three planes, and Bol quasifields, J. Algebra 18 (1971), 159–178.Google Scholar
11. Luneburg, H., Charakterisierungen der endlichen desarguesschen projektiven Ebenen, Math. Z. 85 (1964), 419–450.Google Scholar
12. Luneburg, H., Die Suzukigruppen und ihre geometrien, Lecture Notes in Mathematics (Springer-Verlag, Berlin-Heidelberg-New York, 1965).Google Scholar
13. Ostrom, T. G., Linear transformations and collineations of translation planes, J. Algebra 14 (1970), 405–416.Google Scholar
14. Ostrom, T. G., A class of translation planes admitting elations which are not translations, Arch. Math, 21 (1970), 214–217.Google Scholar
15. Ostrom, T. G., Finite translation planes, Lecture Notes in Mathematics (Springer-Verlag, Berlin- Heidelberg, 1970).Google Scholar
16. Ostrom, T. G., Homologies in translation planes, Proc. London Math. Soc. 26 (1973), 605–629.Google Scholar
17. Ostrom, T. G., Elations infinite translation planes of characteristic 3, Abh. Math. Sem. Hamb. 41 (1974), 179–184.Google Scholar
18. Suzuki, M., On a class of doubly transitive groups, Ann. Math. 75 (1962), 105–145.Google Scholar