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.
| Published in | Pure and Applied Mathematics Journal (Volume 6, Issue 1) |
| DOI | 10.11648/j.pamj.20170601.11 |
| Page(s) | 1-4 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Direct Product, Symmetric Group, Action, Rank, Subdegrees, Cartesian Product, Suborbit
| [1] | Akbas, M. (2001). Suborbital graphs for modular group. Bulletin of the London Mathematical Society. 33:647–652. |
| [2] | Cameron, P. J. (1973). Primitive groups with most suborbits doubly transitive. Geometriae Dedicata 1:434–446. |
| [3] | Cameron, P. J., Gewurz, D. A., and Merola, F. (2008). Product action. Discrete Math. 386–394. |
| [4] | Harary, F. (1969). Graph Theory. Addison-Wesley Publishing Company, New York. |
| [5] | Higman, D.G (1970). Characterization of families of rank 3 permutation groups by the subdegrees. Archiv der Mathematik, 21. |
| [6] | Higman, D. G. (1964). Finite permutation groups of rank 3. Math Zeitschriff, 86:145–156. |
| [7] | Krishnamurthy, V. (1985). Combinatorics, Theory and Applications. Affiliated East-West Press Private Limited, New Delhi. |
| [8] | 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. |
| [9] | Rimberia, J. K. (2011). Ranks and Subdegrees of the Symmetric Group Sn Acting on Ordered r− Element Subsets. PhD thesis, Kenyatta University, Nairobi, Kenya. |
| [10] | Rose, J. S. (1978). A Course in Group Theory. Cambridge University Press, Cambridge. |
| [11] | Sims, C. C. (1967). Graphs and finite permutation groups. Mathematische Zeitschrift, 95:76–86. |
| [12] | Wielandt, H. (1964). Finite Permutation Groups. Academic Press New York. |
APA Style
Gikunju David Muriuki, Nyaga Lewis Namu, Rimberia Jane Kagwiria. (2017). 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, 6(1), 1-4. https://doi.org/10.11648/j.pamj.20170601.11
ACS Style
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 Appl. Math. J. 2017, 6(1), 1-4. doi: 10.11648/j.pamj.20170601.11
AMA Style
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 Appl Math J. 2017;6(1):1-4. doi: 10.11648/j.pamj.20170601.11
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.pamj.20170601.11,
author = {Gikunju David Muriuki and Nyaga Lewis Namu and Rimberia Jane Kagwiria},
title = {Ranks, Subdegrees and Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets},
journal = {Pure and Applied Mathematics Journal},
volume = {6},
number = {1},
pages = {1-4},
doi = {10.11648/j.pamj.20170601.11},
url = {https://doi.org/10.11648/j.pamj.20170601.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20170601.11},
abstract = {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.},
year = {2017}
}
TY - JOUR T1 - Ranks, Subdegrees and Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets AU - Gikunju David Muriuki AU - Nyaga Lewis Namu AU - Rimberia Jane Kagwiria Y1 - 2017/02/02 PY - 2017 N1 - https://doi.org/10.11648/j.pamj.20170601.11 DO - 10.11648/j.pamj.20170601.11 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 1 EP - 4 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.20170601.11 AB - 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. VL - 6 IS - 1 ER -
Information
Email: