The study of suborbital graphs is a key area in group theory for it provides a graphical representation of a group action on a set. Moreover, it helps in understanding the combinatorial structures of the action of a group on a set. In this paper, we construct suborbital graphs based on the group action of the direct product of the symmetric group on Cartesian product of three sets through computation of the ranks and subdegrees of the group action. Suborbital graphs are constructed by the use of Sims theorem. The properties of the suborbital graphs are analyzed. In the study it is proven that the rank of the group action of direct product of the symmetric group acting on the Cartesian product of three sets is 8 for all n ≥ 2 and the suborbits are length 1, (n-1), (n-1), (n-1), (n-1)2, (n-1)2, (n-1)2, (n-1)3. We show that the suborbits of the group action are self-paired. Furthermore, it is demostrated that each graph has a girth of 3 for all n > 2 and suborbital graphs of the group action are undirected. It is shown that graphs Γ2 and Γ3 are regular of degree n-1, graphs Γ4, Γ5 and Γ6 of degree (n-1)2 and graph Γ7 is regular of degree (n-1)3. The suborbital graphs Γi(i=1, 2,…, 6) are disconnected, with the number of connected components equal to n2 while suborbital graph Γ7 is connected for all n > 2.
| Published in | American Journal of Applied Mathematics (Volume 12, Issue 5) |
| DOI | 10.11648/j.ajam.20241205.17 |
| Page(s) | 175-182 |
| 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 |
Rank, Subdegrees, Suborbit, Suborbital Graphs
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APA Style
Muriuki, G. D., Namu, N. L. (2024). Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets. American Journal of Applied Mathematics, 12(5), 175-182. https://doi.org/10.11648/j.ajam.20241205.17
ACS Style
Muriuki, G. D.; Namu, N. L. Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets. Am. J. Appl. Math. 2024, 12(5), 175-182. doi: 10.11648/j.ajam.20241205.17
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Download@article{10.11648/j.ajam.20241205.17,
author = {Gikunju David Muriuki and Nyaga Lewis Namu},
title = {Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets
},
journal = {American Journal of Applied Mathematics},
volume = {12},
number = {5},
pages = {175-182},
doi = {10.11648/j.ajam.20241205.17},
url = {https://doi.org/10.11648/j.ajam.20241205.17},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241205.17},
abstract = {The study of suborbital graphs is a key area in group theory for it provides a graphical representation of a group action on a set. Moreover, it helps in understanding the combinatorial structures of the action of a group on a set. In this paper, we construct suborbital graphs based on the group action of the direct product of the symmetric group on Cartesian product of three sets through computation of the ranks and subdegrees of the group action. Suborbital graphs are constructed by the use of Sims theorem. The properties of the suborbital graphs are analyzed. In the study it is proven that the rank of the group action of direct product of the symmetric group acting on the Cartesian product of three sets is 8 for all n ≥ 2 and the suborbits are length 1, (n-1), (n-1), (n-1), (n-1)2, (n-1)2, (n-1)2, (n-1)3. We show that the suborbits of the group action are self-paired. Furthermore, it is demostrated that each graph has a girth of 3 for all n > 2 and suborbital graphs of the group action are undirected. It is shown that graphs Γ2 and Γ3 are regular of degree n-1, graphs Γ4, Γ5 and Γ6 of degree (n-1)2 and graph Γ7 is regular of degree (n-1)3. The suborbital graphs Γi(i=1, 2,…, 6) are disconnected, with the number of connected components equal to n2 while suborbital graph Γ7 is connected for all n > 2.
},
year = {2024}
}
TY - JOUR T1 - 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 Y1 - 2024/10/18 PY - 2024 N1 - https://doi.org/10.11648/j.ajam.20241205.17 DO - 10.11648/j.ajam.20241205.17 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 175 EP - 182 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20241205.17 AB - The study of suborbital graphs is a key area in group theory for it provides a graphical representation of a group action on a set. Moreover, it helps in understanding the combinatorial structures of the action of a group on a set. In this paper, we construct suborbital graphs based on the group action of the direct product of the symmetric group on Cartesian product of three sets through computation of the ranks and subdegrees of the group action. Suborbital graphs are constructed by the use of Sims theorem. The properties of the suborbital graphs are analyzed. In the study it is proven that the rank of the group action of direct product of the symmetric group acting on the Cartesian product of three sets is 8 for all n ≥ 2 and the suborbits are length 1, (n-1), (n-1), (n-1), (n-1)2, (n-1)2, (n-1)2, (n-1)3. We show that the suborbits of the group action are self-paired. Furthermore, it is demostrated that each graph has a girth of 3 for all n > 2 and suborbital graphs of the group action are undirected. It is shown that graphs Γ2 and Γ3 are regular of degree n-1, graphs Γ4, Γ5 and Γ6 of degree (n-1)2 and graph Γ7 is regular of degree (n-1)3. The suborbital graphs Γi(i=1, 2,…, 6) are disconnected, with the number of connected components equal to n2 while suborbital graph Γ7 is connected for all n > 2. VL - 12 IS - 5 ER -
Department of Mathematical Sciences, The Co-operative University of Kenya, Nairobi, Kenya
Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya
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