Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets

Morang’a Daniel Orina; Nyaga Lewis Namu; Gikunju David Muriuki

doi:doi:10.11648/j.ajam.20241205.16

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Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets

Morang’a Daniel Orina^*

, Nyaga Lewis Namu

, Gikunju David Muriuki

Published in American Journal of Applied Mathematics (Volume 12, Issue 5)

Received: 19 August 2024 Accepted: 6 September 2024 Published: 29 September 2024

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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).

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

Keywords

Ranks, Subdegrees, Transitivity, Primitivity, Direct Product, Cartesian Product, Alternating Group, Cyclic Group

References

[1]	Hungerford, Thomas. W. (1974). Groups. Algebra, First Edition. New York: Springer; 23-69. https://doi.org/10.1007/978-1-4612-6101-8 2
[2]	Kurzweil, H. and Stellmacher, B. (2004). Permutation Groups.The Theory of Finite Groups (pp. 77-97). New York: Springer New York. https://doi.org/10.1007/0-387- 21768-1 4
[3]	Gallian, J. A. (2021). Contemporary Abstract Algebra (10th ed., pp. 83-88). Cengage Learning. https://doi.org/10.1201/9781003142331
[4]	Loh, C. (2017). Group actions. In Geometric Group Theory: An Introduction. Cham: Springer International Publishing; 75-114. https://doi.org/10.1007/978-3-319- 72254-2 4
[5]	Nyaga, L. N., Kamuti, I. N., Mwathi, C. W., Akanga, J. R. (2011). Ranks and Subdegrees of the Symmetric Group SnActing on Unordered r-element Subsets. International Journal of Pure and Applied Mathematics, 3(2), 147-163.
[6]	Denton, T. (2022). Introduction to Algebraic Structures. Fields Institute of York University in Toronto LibreTexts, 54-55.
[7]	Gachimu, R., Kamuti, I., Nyaga, L., Rimberia, J., Kamaku, P. (2016). Properties and Invariants Associated with the Action of the Alternating Group on Unordered Subsets. International Journal of Pure and Applied Mathematics, 106(1), 333-346. https://doi.org/10.12732/ijpam.v106i1.27
[8]	Nyaga, L. N. (2018). Transitivity of the Direct Product of the Alternating Group Acting on the Cartesian Product of Three Sets. International journal of mathematics and its applications, 6(1), 889-893.
[9]	Rose, J. S. (1978). A course on group theory. Cambridge University Press, Cambridge.
[10]	Harary, F. (1969). Graph theory. Addison - Wesley Publishing Company, New York. https://doi.org/10.2307/3615048
[11]	Wambui, M. T., Rimberia, J. (2019). Ranks, Subdegrees, Suborbital graphs and Cycle Indice Associated with the Product Action of A_n × A_n × A_n (4 6 n 6 8) on Cartesian Product of X × Y × Z. Kenyatta University Institutional Repository.
[12]	Mwai, V. (2016). Suborbital Graphs Corresponding to the Action of Dihedral Group and Cyclic Group on the Vertices of a Regular Polygon. JomoKenyatta University of Agriculture and Technology.
[13]	Gustavo, M. L. (2013). Direct product of the group. Mathematics Stack Exchange. https://math.stackexchange.com/q/390352
[14]	Gikunju, M. D., Nyaga, N. L., Rimberia, K. J. (2017). Ranks, SubdegreeandSuborbitalGraphofDirectProduct of 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
[15]	Kadedesya, S., Nyaga, L. and Rimberia, J. (2023). On Suborbits and Graphs Associated with Action of Alternating Groups on Cartesian Product of Two Sets. Asian Research Journal of Mathematics, 19(11), 165- 174. https://doi.org/10.9734/arjom/2023/v19i11763

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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

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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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AMA Style

Orina MD, Namu NL, Muriuki GD. 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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@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}
}

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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
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Author Information

Morang’a Daniel Orina

Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

Contact Email

http://orcid.org/0009-0005-9505-7169
Nyaga Lewis Namu

Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

Contact Email

http://orcid.org/0009-0000-0004-9187
Gikunju David Muriuki

Department of Mathematical Sciences, The Co-operative University of Kenya, Nairobi, Kenya

Contact Email

http://orcid.org/0009-0001-0285-6692

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Plain Text BibTeX RIS

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

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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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AMA Style

Orina MD, Namu NL, Muriuki GD. 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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@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}
}

Copy | Download

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  -

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