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Title: Invariants and projective planes
Author: Charnes, C.
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 1992
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In this thesis we study the isomorphism problem for finite projective planes and in particular for translation planes by using newly defined isomorphism invariants for projective planes. We consider two invariants; one which was proposed by J. H. Conway and is applicable to general projective planes, and another invariant defined only for translation planes. The isomorphism problem poses a serious obstacle in investigations of projective planes, as illustrated by the following remarks (contained in a paper by Hall, Swift and Killgrove). 'No satisfactory mechanical way to identify two isomorphic planes exists whether they be presented by a coordinate system or by an incidence matrix. The preparation of such a method is an interesting question.' The approach developed in this thesis provides a partial solution to this problem. We also study the related problem of determining the automorphism group of a projective plane. It turns out that for two-dimensional translation planes of odd order, the methods developed here reduce this problem to a routine calculation of an invariant. We have implemented the above invariants (and a variant) for computation, and used them to study the translation planes of orders: 52, 72, 82, 112 and 172 arising from the families of 8-dimensional ovoids defined by Conway et al and others. As a consequence of this investigation we obtain a number of new translation planes and determine their groups. We have also established previously unknown isomorphisms between certain translation planes occurring in the literature. We have found that the invariants have certain interesting properties and I have formulated a number of conjectures regarding these. The conjectures have been verified for all projective planes considered in this thesis, and we offer some comments regarding possibilities for their proof.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available