Cycle Indices, Subdegrees And Suborbital Graphs Of Psl(2,Q) Acting On The Cosets Of Some Of Its Subgroups

ABSTRACT

The action of Projective Special Linear group PSL(2; q) on the cosets of

its subgroups is studied. Primitive permutation representations of PSL(2; q)

have been previously studied by Tchuda (1986), Bon and Cohen (1989) and

Kamuti (1992). In particular, the permutation representations on the cosets

of Cq􀀀1

k

; Cq+1

k

; Pq; A4 and D2(q􀀀1)

k

are studied. In the case where it was

previously done, we employ a dierent method or otherwise quote the results

for completeness purpose. Thus, this thesis deals with determination of disjoint

cycle structures, cycle indices, ranks and subdegrees when PSL(2; q) acts on the

cosets of its ve subgroups mentioned above. Cycle indices are obtained using

a method coined by Kamuti (1992), while ranks and subdegrees are determined

using two methods, either algebraic arguments, use of table of marks, a method

proposed by Ivanov et al. (1983) or both. Subdegrees of PSL(2; q) acting on

the cosets of its cyclic subgroup Cq􀀀1

k

are shown to be 1(2); (q 􀀀 1)(q+2) when

q is even and 1(2); ( q􀀀1

2 )(2q+4) when q is odd using reduced pair group action.

In this representation, the number of self-paired suborbits is determined to be

q + 2 when p = 2, q + 3 when q 1 (mod 4) and q + 1 when q 􀀀1 (mod 4).

The suborbit (1;1) is shown to be paired with (0;1). A method for the

construction of suborbital graphs corresponding to the action PSL(2; q) on the

cosets of Cq􀀀1 is given. The constructed suborbital graphs that are directed are shown to be of girth three.