Cycle Indices, Subdegrees And Suborbital Graphs Of PGL (2,q) Acting On The Cosets Of Its Subgroups

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ABSTRACT

The action of and on the cosets of their subgroups is a very active area in enumerative combinatorics. Most researchers have concentrated on the action of these groups on the cosets of their maximal subgroups. For instance Tchuda computed the subdegrees of the primitive permutation representations of . Kamuti determined the subdegrees of primitive permutation representations of . He also constructed suborbital graphs corresponding to the action of on the cosets of However many properties of the action of on the cosets of its subgroups are still unknown. This research is mainly set to investigate the action of on the cosets some of its subgroups namely; and

. Corresponding to each action the disjoint cycle structures, cycle index formulas, ranks and the subdegrees are computed. To obtain cycle index formulas we use a method devised by Kamuti and for the subdegrees and the ranks we use a method proposed by Ivanov et al. which uses marks of a permutation group. For the action of on the cosets of the subdegrees are shown to be

and

and the rank is . For the subdegrees are

and

and the rank is . Suborbital graphs for acting on the cosets of are constructed and their properties analysed. We have established that the number of self paired suborbits is and the paired suborbits are 2. Also suborbital graphs corresponding to suborbits whose elements intersect at a singleton have been shown to be of girth 3. Suborbital graph corresponding to the suborbit containing is found to be of girth 0. Finally suborbital graph corresponding to suborbit with representative of the form

is shown to be of girth 4.

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Cycle Indices, Subdegrees And Suborbital Graphs Of PGL (2,q) Acting On The Cosets Of Its Subgroups

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