Certain automorphisms of free rank planes

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.