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Translation Planes of Dimension Two with Odd Characteristic

Published online by Cambridge University Press:  20 November 2018

T. G. Ostrom
T. G. Ostrom*
Affiliation:
Washington State University, Pullman, Washington
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A translation plane of dimension d over its kernel K = GF(q) can be represented by a vector space of dimension 2d over K. The lines through the zero vector form a “spread”; i.e., a class of mutually independent vector spaces of dimension d which cover the vector space.

The case where d = 2 has aroused the most interest. The more exotic translation planes tend to be of dimension two; a spread in this case can be interpreted as a class of mutually skew lines in projective three-space.

The stabilizer of the zero vector in the group of collineations is a group of semi-linear transformations and is called the translation complement. The subgroup consisting of linear transformations is the linear translation complement.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 5 , 01 October 1980 , pp. 1114 - 1125
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Dixon, J. D., The structure of linear groups (Van Nostrand Reinhold, New York, 1971).Google Scholar
2. Foulser, D. A., Baer p-elements in translation planes, J. Alg. 31 (1974), 354–366.Google Scholar
3. Hering, C., A new class of quasifields, Math Z. 118 (1970), 56–57.Google Scholar
4. Hering, C., On shears of translation planes, Abh. Math. Sem. Hamb. 37 (1972), 258–268.Google Scholar
5. Johnson, N. L. and Ostrom, T. G., Translation planes of characteristic two in which all involutions are Baer, J. Alg. 54 (1978), 291–315.Google Scholar
6. Johnson, N. L. and Ostrom, T. G., Translation planes of dimension two and characteristic two, submitted.Google Scholar
7. Ostrom, T. G., Linear transformations and collineations of translation planes, J. Alg. 14 (1970), 405–416.Google Scholar
8. Ostrom, T. G., Collineation groups whose order is prime to the characteristic, Math. Z. 156 (1977), 59–71.Google Scholar
9. Schaeffer, H.-J., Translationsebenen auf denen die Gruppe SL(2, pn) operiert, Diplomarbeit Universitàt Tubingen (1975).Google Scholar
10. Suprunenko, I. D. and Zalesskii, A. E., Classification of finite irreducible linear groups of degree 4 on fields of characteristic p > 5, Inst. Mat. Akad. Nauk BSSR Preprint No. B (1976).Google Scholar
11. Walker, M., On translation planes and their collineation groups, Thesis, Westfield College, University of London (1973).Google Scholar
12. Wong, W. J., Representations of Chevalley groups in characteristic p, Nagoya Math. J. 45 (1971), 37–78.Google Scholar