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Regular Orbits of Collineation Groups

Regular Orbits of Collineation Groups ALAN R. PRINCE The purpose of this paper is to prove the following theorem. THEOREM. Let G be a finite group of collineations of a finite projective plane & of order n. Let the number of regular point orbits be k. Then n<\G\ + V\G\(\G\+k). Proof. Let / denote the permutation character of G in its action on the points of the plane. Then the number N of orbits of G in this action is given by the following formula, due to Frobenius: . An orbit is regular if its length is |G|. An orbit which is not regular has length ThuS = ±\G\(N+k). (1) If geG, the fixed points and fixed lines of g form a substructure of & [1]. If g does not fix a subplane of ^ , then all its fixed points, except for perhaps one, lie on a line. Thus x(g) ^n + 2. If g ^ 1 and g fixes a subplane of order m, then by a theorem of 2 2 2 R. H. Bruck (see [2]) either n = m or « ^ m + m. In either event, n ^ m . Thus Xig) < « + V«+1 - Clearly, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Regular Orbits of Collineation Groups

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/20.2.129
Publisher site
See Article on Publisher Site

Abstract

ALAN R. PRINCE The purpose of this paper is to prove the following theorem. THEOREM. Let G be a finite group of collineations of a finite projective plane & of order n. Let the number of regular point orbits be k. Then n<\G\ + V\G\(\G\+k). Proof. Let / denote the permutation character of G in its action on the points of the plane. Then the number N of orbits of G in this action is given by the following formula, due to Frobenius: . An orbit is regular if its length is |G|. An orbit which is not regular has length ThuS = ±\G\(N+k). (1) If geG, the fixed points and fixed lines of g form a substructure of & [1]. If g does not fix a subplane of ^ , then all its fixed points, except for perhaps one, lie on a line. Thus x(g) ^n + 2. If g ^ 1 and g fixes a subplane of order m, then by a theorem of 2 2 2 R. H. Bruck (see [2]) either n = m or « ^ m + m. In either event, n ^ m . Thus Xig) < « + V«+1 - Clearly,

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Mar 1, 1988

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