The interaction between geometry and algebra is a diverse and fruitful area to explore. Of particular interest is the area of groups and geometry. In this work we will examine various aspects of this interplay, in particular looking at geometric canonical forms of figures in various projective spaces under the action of the (projective) general linear group, and finding associated canonical forms for polynomial functions. We deal with some quite complex cases, in particular with the case of line (complement) generated figures of , where it was originally uncertain if the results would be forthcoming. During this work we develop some new techniques for examining these orbit structures, and will see how the use of computers can be very helpful, especially the use of symbolic computer packages. As we make our journey we will see some of the links of these projective spaces with other areas, especially with boolean algebras and exterior algebras, and at how our results can determine various designs and codes. We also look at an example of how finite geometry relates to the algebra of Octonions, and finally we will see how a set of polynomial equations characterizing a set of elements can be reduced to a single polynomial equation which also characterizes the set (for example the set of decomposable bivectors of V).
(1994). Finite Geometry and Computer Algebra, with Applications. (Thesis). Retrieved from https://hull-repository.worktribe.com/output/405456