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On Finite Collineation Groups of F5

Published online by Cambridge University Press:  20 November 2018

Reuben Sandler
Reuben Sandler*
Affiliation:
University of Illinois at Chicago Circle, Chicago, Illinois
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Our aim in this paper is to fill one gap left in (1) and to prove that if H is a finite collineation group of F5, the free plane generated by a finite open configuration of rank 9, then |H | ≦ 12. Alltop has shown that |H | ≦ 24 and that there exist finite collineation groups of F5 which have order 12, so that the argument in this paper shows that |H| ≦ 12 is the best estimate which can be given. In (1), Alltop has completely settled the question for Fn, n ≠ 5. The notation of this paper will generally be that of (1).

Most of the arguments used here will consist of case analyses of degenerate planes of ranks 7 and 8 and will be sketched rather than given in detail. The one theorem of some interest in this paper, other than the main result, is the following theorem which yields some information about the collineation group of F4 (π2 in the notation of (4)).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 217 - 221
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Alltop, W. O., Free planes and collineations, Can. J. Math. 20 (1968), 1397–1411.Google Scholar
2. Marshall, Hall, Jr., Projective planes, Trans. Amer. Math. Soc. 54 (1943), 229–277.Google Scholar
3. Gunter, Pickert, Projective Ebenen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berucksichtigung der Anwendungsgebiete, Bd, LXXX (Springer-Verlag, Berlin, 1955).Google Scholar
4. Reuben, Sandler, The collineation groups of free planes, Trans. Amer. Math. Soc. 107 (1963), 129–139.Google Scholar