Action of GL(2,q) and GL(3,q) on Non Zero Vectors Over GF(q)

Abstract

The action of the General Linear group has been studied by several researchers. Most of them concentrated in deriving the cycle index formula of GL(n,q) leaving out combinatorial properties, invariants and structures of this group. This thesis determines transitivity, primitivity, ranks, subdegrees and suborbital graphs of the action of GL(2,q) and GL(3,q) on their non zero vectors over Fq. In this study, Orbit-Stabilizer theorem was used to determine transitivity and it was found that, both GL(2,q) and GL(3,q) act transitively on F2 q \{0}and F3 q \{0}respectively. The rank of GL(2,q) acting on F2 q \{0}is q while the subdegrees are [1][q−1] and q2−q. In the action of GL(3,q) on F3 q \{0}, rank is q while the subdegrees are [1][q−1] and q3−q. The suborbital graphs of these two actions were constructed using Sims procedure. It was observed that all suborbital graphs corresponding to suborbit of length q2−q in the action of GL(2,q) on F2 q \{0} are: connected, undirected, regular and completefor q=2. Thediameteris1where, q=2and2for q>2. Alsointhe sameaction, the suborbital graphs corresponding to suborbits of length 1 are: regular, disconnected and have chromatic number as either 2 or 3. In the action of GL(3,q) on F3 q \{0}the suborbital graphs corresponding to suborbits of q3−q are connected, self-paired and complete for q = 2. Also in the same action, Γi corresponding to ∆i where|∆i|= 1 is disconnected, regular and diameter is ∞. In conclusion, primitivity was determined using both the graphical method and the stabilizer as maximal subgroup approach. It was ascertained that both actions were primitive where q = 2 and imprimitive where q≥3.