Ranks, Subdegrees and Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets

Gikunju David Muriuki; Nyaga Lewis Namu; Rimberia Jane Kagwiria

doi:doi:10.11648/j.pamj.20170601.11

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Ranks, Subdegrees and Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets

Gikunju David Muriuki, Nyaga Lewis Namu, Rimberia Jane Kagwiria

Published in Pure and Applied Mathematics Journal (Volume 6, Issue 1)

Received: 17 December 2016 Accepted: 3 January 2017 Published: 2 February 2017

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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.

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), 2017. Published by Science Publishing Group

Keywords

Direct Product, Symmetric Group, Action, Rank, Subdegrees, Cartesian Product, Suborbit

References

[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.

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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

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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

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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

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@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}
}

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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
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VL  - 6
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Author Information

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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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

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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

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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

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@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}
}

Copy | Download

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  -

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