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Abstract

Projective planes of finite free rank are planes freely
generated by openly finite configurations (see Hughes and Piper, 1972,
Chapter XI). We use the concept of a hyperfree extension process to
obtain some properties of the finite collineation groups and polarities
of such planes.
We first obtain some basic properties of projective planes, free
completions, hyperfree extension processes and free rank planes, together
with some properties useful for our investigation.
The main work of the thesis is concerned with finite collineation
groups which fix elementwise the confined core of a plane of finite
free rank. Most of the known properties of such groups are obtained,
as well as some which, as far as is known to the author, have not
previously been obtained. If G is such a group, we determine I G |
when G is cyclic~ we obtain upper bounds for both I G | and the number of
conjugacy classes to which G can belong, and we investigate the subplane
of elements fixed by G. As our basic tool, we use the existence of a
hyperfree completion process Q for the plane from its confined core,
such that each configuration of Q is invariant under G.
We then use similar methods to prove most known results about
polarities of planes of finite free rank. Finally, we consider planes
not having free rank, such as open, non-free planes. We give a
generalization of a theorem of Kopejkina and use it to prove a theorem
about some collineation groups of such planes.

Item Type:	Thesis - Coursework Master
Authors/Creators:	Kelly, Graham S
Keywords:	Automorphisms, Finite groups
Copyright Holders:	The Author
Copyright Information:	Copyright 1977 the Author - The University is continuing to endeavour to trace the copyright owner(s) and in the meantime this item has been reproduced here in good faith. We would be pleased to hear from the copyright owner(s).
Additional Information:	Thesis (M.Sc.)--University of Tasmania, 1977. Bibliography: l. 169-171
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