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.
