Finite Translation Planes
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158
T. G. Ostrom Washington State University, PullmanlWAIUSA
Finite Translation Planes
Springer-Verlag Berlin· Heidelberg · NewYork 1970
These lectures are based on research which was supported in part by N. S. F. Grants GP 5036 and GP 7921.
ISBN 3-540-05186-4 ISBN 0-387-05186-4 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § '4 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1970. LibraryofCongress CatalogCard Number 730139232 Printed in Germany.
TABLE OF CONTENTS
General introduction.......................................... Book I
1
Some translation planes that are not well known
1.
Introduction, Book I.................................
4
2.
Preliminary results..................................
4
3.
Constructions by replaceable nets....................
8
4.
(a)
The first method of choosing replaceable nets ••
10
(b)
The second method ••••••••••••••••••••••••••••••
10
(c)
The third method •••••••••••••••••••••••••••••••
11
(d)
The first type of replacement ••••••••••••••••••
13
(e)
The second type ..........................................
15
( f)
The third type ..........................................
17
( g)
The fourth type .........................................
18
Collineations and isomorphisms •••••••••••••••••••••••
21
(a)
Desarguesian decompositions ••••••••••••••••••••
21
(b)
Desarguesian decompositions and collineations of generalized Andre' planes •••••••••••••••••••
27
(c)
Further results on Desarguesian decompositions
and collinea tiona. ...... .... .... ...... .. .. .......... .. .... ..... ...... ......
5.
Tentative classification by Desarguesian
6.
Appendix
Book II
decompos1 tiona
; .. ..
37 41
(Examples) •••••••••••••••••••••••••••.•••••
45
Linear groups and collineation groups ••••••••••••••••
49
Chapter I
Introduction. . . . . . . • . • . . . . . . . . • . • . . . . . . . . . . • •.
49
Chapter II
Desarguesian decompositions •••••••••••••••••••
56
Chapter III
Homologies (general resul ts) •• • •• • • • • • •• • • • • ••
61
Chapter IV
Homologies as linear transformations..........
69
(a) (b) Chapter V Chapter VI
Collineation groups containing
homologies. . . • • •• . • •. .. . •• • •. . .•. •.• ••• •
69
An unusual plane of order 38 •
80
• • • • • • • • • • • •
Homologies and generalized Andre' planes...... 83 Translation planes admitting elations which
are not transla tiona. . . . . . . . . . . • . . . . . .. . . • • • ••
91
IV (a)
Appendix
SL(2,p·) •••
91
The group generated by elations •••••••••
94
A class of planes admitting
Known finite translation planes and their properties •••••••
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