COUNTING G-ORBITS ON THE INDUCED ACTION ON k-SUBSETS

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Abstract

Let G be a finite permutation group acting on a finite set Ω. Then we denote by σk(G,Ω) the number of G-orbits on the set Ωk, consisting of all k-subsets of Ω. In this thesis we develop methods for calculating the values for σk(G,Ω) and produce formulae for the cases that G is a doubly-transitive simple rank one Lie type group. That is G ∼ = PSL(2,q),Sz(q),PSU(3,q) or R(q). We also give reduced functions for the calculation of the number of orbits of these groups when k = 3 and go on to consider the numbers of orbits, when G is a finite abelian group in its regular representation.We then consider orbit lengths and examine groups with “large” G-orbits on subsetsof size 3

Keyword(s)

Algebra, Group Theory, subsets, orbits, counting, combinatorics, finite simple groups, 2-transitive

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