Ranks, Subdegrees and Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets
Pure and Applied Mathematics Journal
Volume 6, Issue 1, February 2017, Pages: 1-4
Received: Dec. 17, 2016;
Accepted: Jan. 3, 2017;
Published: Feb. 2, 2017
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Gikunju David Muriuki, Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya
Nyaga Lewis Namu, Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya
Rimberia Jane Kagwiria, Department of Pure and Applied Sciences, Kenyatta University, Nairobi, Kenya
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Transitivity and Primitivity of the action of the direct product of the symmetric group on Cartesian product of three sets are investigated in this paper. We prove that this action is both transitive and imprimitive for all n ≥ 2. In addition, we establish that the rank associated with the action is a constant 23 Further; we calculate the subdegrees associated with the action and arrange them according to their increasing magnitude.
Direct Product, Symmetric Group, Action, Rank, Subdegrees, Cartesian Product, Suborbit
To cite this article
Gikunju David Muriuki,
Nyaga Lewis Namu,
Rimberia Jane Kagwiria,
Ranks, Subdegrees and Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets, Pure and Applied Mathematics Journal.
Vol. 6, No. 1,
2017, pp. 1-4.
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
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Akbas, M. (2001). Suborbital graphs for modular group. Bulletin of the London Mathematical Society. 33:647–652.
Cameron, P. J. (1973). Primitive groups with most suborbits doubly transitive. Geometriae Dedicata 1:434–446.
Cameron, P. J., Gewurz, D. A., and Merola, F. (2008). Product action. Discrete Math. 386–394.
Harary, F. (1969). Graph Theory. Addison-Wesley Publishing Company, New York.
Higman, D.G (1970). Characterization of families of rank 3 permutation groups by the subdegrees. Archiv der Mathematik, 21.
Higman, D. G. (1964). Finite permutation groups of rank 3. Math Zeitschriff, 86:145–156.
Krishnamurthy, V. (1985). Combinatorics, Theory and Applications. Affiliated East-West Press Private Limited, New Delhi.
Nyaga, L. N. (2012). Ranks, Subdegrees and Suborbital Graphs of the Symmetric Group Sn Acting on Unordered r− Element Subsets. PhD thesis, JKUAT, Nairobi, Kenya.
Rimberia, J. K. (2011). Ranks and Subdegrees of the Symmetric Group Sn Acting on Ordered r− Element Subsets. PhD thesis, Kenyatta University, Nairobi, Kenya.
Rose, J. S. (1978). A Course in Group Theory. Cambridge University Press, Cambridge.
Sims, C. C. (1967). Graphs and finite permutation groups. Mathematische Zeitschrift, 95:76–86.
Wielandt, H. (1964). Finite Permutation Groups. Academic Press New York.