In relation to group action, much research has focused on the properties of individual permutation groups acting on both ordered and unordered subsets of a set, particularly within the Alternating group and Cyclic group. However, the action of the direct product of Alternating group and Cyclic group on the Cartesian product of two sets remains largely unexplored, suggesting that some properties of this group action are still undiscovered. This research paper therefore, aims to determine the combinatorial properties - specifically transitivity and primitivity - as well as invariants which includes ranks and subdegrees of this group action. Lemmas, theorems and definitions were utilized to achieve the objectives of study with significant use of the Orbit-Stabilizer theorem and Cauchy-Frobeneus lemma. Therefore in this paper, the results shows that for any value of n ≥ 3, the group action is transitive and imprimitive. Additionally, we found out that when n = 3, the rank is 9 and the corresponding subdegrees are ones repeated nine times that is, 1, 1, 1, 1, 1, 1, 1, 1, 1. Also, for any value of n > 4, the rank is 2n with corresponding subdegrees comprising of n suborbits of size 1 and n suborbits of size (n − 1).
| Published in | American Journal of Applied Mathematics (Volume 12, Issue 5) |
| DOI | 10.11648/j.ajam.20241205.16 |
| Page(s) | 167-174 |
| 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 |
Ranks, Subdegrees, Transitivity, Primitivity, Direct Product, Cartesian Product, Alternating Group, Cyclic Group
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APA Style
Orina, M. D., Namu, N. L., Muriuki, G. D. (2024). Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets. American Journal of Applied Mathematics, 12(5), 167-174. https://doi.org/10.11648/j.ajam.20241205.16
ACS Style
Orina, M. D.; Namu, N. L.; Muriuki, G. D. Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets. Am. J. Appl. Math. 2024, 12(5), 167-174. doi: 10.11648/j.ajam.20241205.16
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Download@article{10.11648/j.ajam.20241205.16,
author = {Morang’a Daniel Orina and Nyaga Lewis Namu and Gikunju David Muriuki},
title = {Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets},
journal = {American Journal of Applied Mathematics},
volume = {12},
number = {5},
pages = {167-174},
doi = {10.11648/j.ajam.20241205.16},
url = {https://doi.org/10.11648/j.ajam.20241205.16},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241205.16},
abstract = {In relation to group action, much research has focused on the properties of individual permutation groups acting on both ordered and unordered subsets of a set, particularly within the Alternating group and Cyclic group. However, the action of the direct product of Alternating group and Cyclic group on the Cartesian product of two sets remains largely unexplored, suggesting that some properties of this group action are still undiscovered. This research paper therefore, aims to determine the combinatorial properties - specifically transitivity and primitivity - as well as invariants which includes ranks and subdegrees of this group action. Lemmas, theorems and definitions were utilized to achieve the objectives of study with significant use of the Orbit-Stabilizer theorem and Cauchy-Frobeneus lemma. Therefore in this paper, the results shows that for any value of n ≥ 3, the group action is transitive and imprimitive. Additionally, we found out that when n = 3, the rank is 9 and the corresponding subdegrees are ones repeated nine times that is, 1, 1, 1, 1, 1, 1, 1, 1, 1. Also, for any value of n > 4, the rank is 2n with corresponding subdegrees comprising of n suborbits of size 1 and n suborbits of size (n − 1).},
year = {2024}
}
TY - JOUR T1 - Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets AU - Morang’a Daniel Orina AU - Nyaga Lewis Namu AU - Gikunju David Muriuki Y1 - 2024/09/29 PY - 2024 N1 - https://doi.org/10.11648/j.ajam.20241205.16 DO - 10.11648/j.ajam.20241205.16 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 167 EP - 174 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20241205.16 AB - In relation to group action, much research has focused on the properties of individual permutation groups acting on both ordered and unordered subsets of a set, particularly within the Alternating group and Cyclic group. However, the action of the direct product of Alternating group and Cyclic group on the Cartesian product of two sets remains largely unexplored, suggesting that some properties of this group action are still undiscovered. This research paper therefore, aims to determine the combinatorial properties - specifically transitivity and primitivity - as well as invariants which includes ranks and subdegrees of this group action. Lemmas, theorems and definitions were utilized to achieve the objectives of study with significant use of the Orbit-Stabilizer theorem and Cauchy-Frobeneus lemma. Therefore in this paper, the results shows that for any value of n ≥ 3, the group action is transitive and imprimitive. Additionally, we found out that when n = 3, the rank is 9 and the corresponding subdegrees are ones repeated nine times that is, 1, 1, 1, 1, 1, 1, 1, 1, 1. Also, for any value of n > 4, the rank is 2n with corresponding subdegrees comprising of n suborbits of size 1 and n suborbits of size (n − 1). VL - 12 IS - 5 ER -
Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya
Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya
Department of Mathematical Sciences, The Co-operative University of Kenya, Nairobi, Kenya
