Classifying Flag-Transitive Linear Spaces using Cyclic Line-Spreads and Orbit Polynomials

In this thesis we are concerned with linear spaces that have the property of flag-transitivity. Much progress has been made on the problem of classifying such linear spaces, with a seminal result by Buekenhout et al. in 1990 leaving only a few cases unsolved. In 2008, Pauley and Bamberg provided a polynomial condition for when we can attain a flag-transitive linear space in one of the open cases. We aim to determine which polynomials can satisfy this condition. In Chapter 1, we introduce the problem in more detail and review its background. We also reformulate the condition of Pauley and Bamberg into a question involving a curve and provide some initial results on this curve. We focus on the cubic case in Chapter 2. This is the smallest open case and allows us to gain some intuition on how the problem operates in general. We attain a complete characterisation of degree 3 polynomials satisfying the condition. In order to determine when these polynomials give rise to equivalent flag-transitive linear spaces, we spend Chapter 3 developing theory and results that are necessary for later chapters. This involves concepts like the projective linear group, linear fractional transformations and orbit polynomials. With this theory in tow, we classify the equivalence classes of cubic polynomials in Chapter 4 and provide equivalence representatives. We turn to quartic and quintic polynomials in Chapter 5. Unlike the cubic case, a complete classifiction is not obtained for these higher degrees, though we prove some results on existence and construct some examples. Finally, in Chapter 6 we examine two known families of polynomials satisfying the condition of Pauley and Bamberg. We interpret these families in terms of our curve approach and use our methods to extend them to find new inequivalent polynomials.