Ranks, Subdegrees and Suborbital Graphs of Symmetric Group Sn Acting on Ordered Pairs
DOI:
https://doi.org/10.53555/nnas.v3i2.664Keywords:
Ranks, Subdegrees, uborbitals, Suborbital graphs, Ordered pairsAbstract
In this research paper, we study the ranks and subdegrees of the symmetric group Sn (n = 3, 4, 5) acting on ordered pairs from the set X = {1, 2 , … , n}. When Sn (n ? 4) acts on ordered pairs from X, the rank is 7. Therefore the main study will be on the ranks and subdegrees of the suborbitals. The suborbital graphs corresponding to the suborbitals of these actions are also constructed. The graph theoretic properties of these suborbital graphs are also discussed. When Sn (n ? 4) acts on ordered pairs, the suborbital graphs, ?1,?2, ?5, and ?6 corresponding to the non-trivial suborbits, ?1 , ?2 , ?5and ?6 are disconnected, regular and undirected. The suborbital graphs ?3and ?4 are disconnected, and directed.
References
Akbas M (2001). Suborbital graphs for Modular group, Bulletin of the London Mathematical Society 33:647 –652.
Cameron PJ (1972). Permutation groups with multiply transitive suborbits I. Proc. LondonMath.Soc. 23(3):427 – 440.
Cameron PJ (1974). Permutation groups with multiply transitive suborbits II. Bull. LondonMath.Soc.6:136 –140.
Cameron PJ (1978). Orbits of permutation groups on unordered sets. J. LondonMath.Soc.17(2):410 – 414.
Chartrand G (1993). Applied and algorithmic graph theory. International series in pure and applied mathematics: 238-240.
Coxeter HSM (1986). My graph, Proceedings of London Mathematical Society 46: 117 – 135.
Harary F (1969). Graph Theory. Addison-Wesley Publishing Company New York.
Higman DG (1970). Characterization of families of rank 3 permutation groups by subdegrees. I. Arch. Math21: 151-156.
Kamuti IN (1992). Combinatorial formulas, invariants and structures associated with primitive permutation representations of PSL (2, q) and PGL (2, q). Ph. D. Thesis, University of Southampton, U.K.
Kamuti IN (2006). Subdegrees of primitive permutation representation of, PGL(2,q) , East African Journal of Physical Sciences 7(1/2): 25 – 41.
Kangogo M (2008). Properties of some actions of the symmetric group S6and associated combinatorial formulas and structures, M.Sc. Project; Kenyatta University, Kenya. Krishnamurthy V (1985). Combinatorics, theory and applications. Affliated East West Pressprivate Limited, New Delhi.
Neumann PM (1977). Finite permutation groups Edge – coloured graphs and matrices, edited by M. P. J. Curran, Academic Press, London.
Rotich KS (2008). Some properties of the symmetric group S7acting on ordered and unordered pairs and the associated combinatorial structures, M.Sc. Project, Kenyatta University, Kenya.
Sims CC (1967). Graphs and finite permutation groups. Math. Zeitschrift.95: 76 – 86.
Wielandt H (1964). Finite permutation groups. Academic Press, New York and London.
Downloads
Published
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
You are free to:
- Share — copy and redistribute the material in any medium or format for any purpose, even commercially.
- Adapt — remix, transform, and build upon the material for any purpose, even commercially.
- The licensor cannot revoke these freedoms as long as you follow the license terms.
Under the following terms:
- Attribution — You must give appropriate credit , provide a link to the license, and indicate if changes were made . You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
Notices:
You do not have to comply with the license for elements of the material in the public domain or where your use is permitted by an applicable exception or limitation .
No warranties are given. The license may not give you all of the permissions necessary for your intended use. For example, other rights such as publicity, privacy, or moral rights may limit how you use the material.
