Combinatorial Properties, Invariants, Structures and Formulas Associated with Some Actions of the Alternating Group

Abstract

Various group actions have been studied in the past with respect to their associated combinatorial properties, invariants, structures and formulas. This thesis focuses on the combinatorial properties, invariants and structures of the alternating group An acting on X[r] and X(r), respectively the ordered and unordered r-element subsets of the set X of n letters. It also aims at deriving an expression of the cycle index of the symmetric group Sn, a semidirect product of An by the cyclic group C2 of order 2, explicitly in terms of the cycle index of An and that of C2. Transitivity of the actions is established using either the Cauchy-Frobenius Lemma or the Orbit-Stabilizer Theorem; primitivity is determined from the definition of blocks; ranks and subdegrees are computed using combinatorial arguments; pairing of suborbits is determined from definition; the suborbital graphs are constructed from their corresponding suborbitals; and the cycle index is derived from definition. The study shows that the action of An on X[r] is transitive and imprimitive if and only if n ≥ r + 2, while the rank is constant for all n ≥ 2(r + 1). On the other hand, the action of An on X(r) is shown to be transitive for all n ≥ r + 1 and imprimitive if and only if n = 2r, while the rank is r + 1 if and only if n ≥ 2r. Further, the ranks and subdegrees of the two actions are calculated and pairing of the associated suborbits explored. Moreover, suborbital graphs related to the actions are seperately constructed and examined for directedness, connectedness, number of components, vertex degrees, and girths, depending on whether a corresponding suborbit is self-paired or paired with another, and also the number of elements from a fixed r-element subset that each element of the suborbit has. Finally, an expression of the cycle index of Sn, explicitly in terms of the cycle index of An and that of C2, is obtained.