The action of projective general group on the cosets of its maximal subgroups has been studied. For instance, [9] studied the action of G on the cosets of P GL(2, e) when q is an odd prime power of e. In this paper, we determine the rank and subdegrees of the action of P GL(2, q) on the cosets of its subgroup P GL(2, e) for odd q and an even power of e. We apply the table of marks to achieve this.
The transitivity, primitivity, rank and subdegrees, as well as pairing of the suborbits associated with the action of the alternating group An, on unordered r−element subsets of a set X = {1, 2, • • • , n} of n letters, have not received any attention. In this paper, we prove that this action is transitive. We also show that the action is imprimitive if and only if n = 2r. In addition, we establish that the rank associated with the action is a constant r + 1 if and only if n ≥ 2r, except for r = 2 in which case the rank is 4 if n = 4, but is 3 for all n ≥ 5. Further, we calculate the subdegrees associated with the action and arrange them according to their increasing magnitudes. Finally, we show that all the suborbits of the action, with the exception of some non-trivial suborbits corresponding to the actions of A3 and A4 on the set of unordered pairs, are self-paired.
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