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Involutory homologies on affine planes

Involutory homologies on affine planes Abh. Math. Sem. Univ. Hamburg 57, 37-55 (1986) By U. DEMPWOLFF Dedicated Prof. B. HUPPERT to this sixtieth birthday. 1. Introduction. Let ~ = (~, 5f) be a projective plane (~ = set of points, ~ = set of lines) and let G be a finite subgroup of Aut (~) fixing a nonincident point-line pair (Z, 1). Assume G is generated by a normal set R ofinvolutory homologies. As usual G (P) denotes the set of perspectivities in G with center P, G (g) denotes the set of perspectivities in G with axis g, and G (P, g) = G (P) ~ G (g). In view of applications (i.e. translation planes) we make the following assumption: (P 1). Let (P, g) be a nonincident point-line pair and S ~ Syl 2 (G (P, g)), then S con- tains at most one involution. In section 2 we give an obvious group theoretic consequence of (P1)-namely (P2)- and first reductions. In sections 3 and 4 we treat some special cases, which are used in the general case in section 5. The classification of finite simple groups is the essential tool to obtain the main results 5.3 and 5.4. 2. Preliminary reductions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

Involutory homologies on affine planes

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02941598
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Abstract

Abh. Math. Sem. Univ. Hamburg 57, 37-55 (1986) By U. DEMPWOLFF Dedicated Prof. B. HUPPERT to this sixtieth birthday. 1. Introduction. Let ~ = (~, 5f) be a projective plane (~ = set of points, ~ = set of lines) and let G be a finite subgroup of Aut (~) fixing a nonincident point-line pair (Z, 1). Assume G is generated by a normal set R ofinvolutory homologies. As usual G (P) denotes the set of perspectivities in G with center P, G (g) denotes the set of perspectivities in G with axis g, and G (P, g) = G (P) ~ G (g). In view of applications (i.e. translation planes) we make the following assumption: (P 1). Let (P, g) be a nonincident point-line pair and S ~ Syl 2 (G (P, g)), then S con- tains at most one involution. In section 2 we give an obvious group theoretic consequence of (P1)-namely (P2)- and first reductions. In sections 3 and 4 we treat some special cases, which are used in the general case in section 5. The classification of finite simple groups is the essential tool to obtain the main results 5.3 and 5.4. 2. Preliminary reductions.

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Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Dec 1, 1987

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