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

John Mokaya Victor; Nyaga Lewis Namu; Gikunju David Muriuki

doi:doi:10.11648/j.ajam.20241206.15

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

John Mokaya Victor^*

, Nyaga Lewis Namu

, Gikunju David Muriuki

Published in American Journal of Applied Mathematics (Volume 12, Issue 6)

Received: 6 November 2024 Accepted: 19 November 2024 Published: 18 December 2024

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Abstract

Group theory, an area in mathematics, has undergone extensive research, but not exhaustively. In group theory, especially symmetric groups have been studied in terms of properties and actions. In this research, two groups of the Symmetric Group (Alternating and Dihedral Groups) are studied in terms of their properties and also their direct product between them. Also, ordered sets are studied in this research where their Cartesian product is looked at. Finally, this research combines the direct product of two symmetric groups (alternating group and dihedral group) and the Cartesian product of two ordered sets (X×Y) by group action. This research therefore focuses on determining the combinatorial properties (transitivity and primitivity), invariants (ranks and subdegrees), and structures (suborbital graphs) of this group action. To accomplish this, orbit-stabilizer theorem is used to compute transitivity, block action concept is applied to determine primitivity, and Cauchy-Frobenius lemma is used to compute ranks and subdegrees. For n ≥ 3, it has been confirmed that the group action is transitive and also imprimitive. The rank of this group action is 6 and the subdegrees are obtained using the formula of Theorem 3.3. This research adds new concepts in group theory which will be useful in other areas like graph theory and also application in real life situations.

Published in	American Journal of Applied Mathematics (Volume 12, Issue 6)
DOI	10.11648/j.ajam.20241206.15
Page(s)	258-265
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

Group Action, Ranks, Subdegrees, Transitivity, Primitivity, Alternating Group, Dihedral Group

References

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[2]	Kurzweil, H. and Stellmacher, B. (2004). Permutation Groups. The Theory of FiniteGroups (pp. 77-97). New York: Springer New York. https://doi.org/10.1007/0-87-21768-14
[3]	Burnside William. (1911). Theory of Groups of finite Order. Cambridge University Press. 2.1. https://doi.org/10.1017/CBO9781139237253.023
[4]	Thomas, W. J. (2020). Abstract Algebra (Theory and Applications). Stephen F. Austin University. https://scholarworks.sfasu.edu/ebooks/23
[5]	abstractGroups2014 Abstract Groups. (2014). A First Course in Abstract Algebra, 224–235. https://doi.org/10.1201/b17673-28
[6]	Gachogu, R., Kamuti, I. N., Gachuki, M. N. (2017). Properties of Suborbitals of Dihedral group acting on ordered subset. Advances in Pure Mathematics, 7, 375- 382. https://doi.org/10.4236/apm.2017.78024
[7]	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
[8]	Gachimu, R., kamuti, I., Nyaga, L., Rimberia, J., Kamaku, P. (2016).Properties 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
[9]	Njagi, L. (2016). Ranks and Subdegrees of Suborbital Graphs of Symmetric Group Acting on Ordered Pairs. Journal of Advanced Research in Applied Science, 3(2). https://doi.org/10.53555/nnas.v3i2.664
[10]	Siro, M., Kamuti, I, Maingi, M. (2014). Actions of Symmetric Group Sn(n 6 7) on Unordered Quadruples. International Journal of Algebra, 8(3),115- 120. https://doi.org/10.12988/ija.2014.311112
[11]	Gikunju, M. D., Nyaga, N. L., Rimberia, K. J. (2017). Ranks, Subdegree and Suborbital Graph of Direct Product 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
[12]	Scott, W. R. (1987). Group Theory.Dover Publications Inc. New York. Second Edition. Page 2.
[13]	Joseph, A. G. (2012). Contemporary Abstract ALgebra.Brooks Cole Cengage Learning. 8th Edition.
[14]	Gustavo, M. L. (2013). Direct product of the group. Mathematics Stack Exchange. https://math.stackexchange.com/q/390352
[15]	Rose, J. S. (1978). A course on group theory. Cambridge University Press, Cambridge.
[16]	Harary, F. (1969). Graph theory. Addison - Wesley Publishing Company, New York. https://doi.org/10.2307/3615048
[17]	Wambui, M. T., Rimberia, J. (2019). Ranks, Subdegrees, Suborbital Graphs and Cycle Indices Associated with the Product Action of An × An × An(4 6 n 6 8) on Cartesian Product X × Y × Z. Kenyatta University Institutional Repository. http://ir-library.ku.ac.ke/handle/123456789/20050
[18]	Ivanov, A. A (1983). On The Problem of Computing Subdegrees of Transitive Permutation Groups. Soviet Mathemetical Survey. 38. 123-124. https://doi.org/10.1070/rm1983v038n06abeh003460

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APA Style

Victor, J. M., Namu, N. L., Muriuki, G. D. (2024). Combinatorial Properties and Invariants Associated with the Direct Product of Alternating and Dihedral Groups Acting on the Cartesian Product of Two Sets. American Journal of Applied Mathematics, 12(6), 258-265. https://doi.org/10.11648/j.ajam.20241206.15

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ACS Style

Victor, J. M.; Namu, N. L.; Muriuki, G. D. Combinatorial Properties and Invariants Associated with the Direct Product of Alternating and Dihedral Groups Acting on the Cartesian Product of Two Sets. Am. J. Appl. Math. 2024, 12(6), 258-265. doi: 10.11648/j.ajam.20241206.15

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AMA Style

Victor JM, Namu NL, Muriuki GD. Combinatorial Properties and Invariants Associated with the Direct Product of Alternating and Dihedral Groups Acting on the Cartesian Product of Two Sets. Am J Appl Math. 2024;12(6):258-265. doi: 10.11648/j.ajam.20241206.15

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@article{10.11648/j.ajam.20241206.15,
  author = {John Mokaya Victor and Nyaga Lewis Namu and Gikunju David Muriuki},
  title = {Combinatorial Properties and Invariants Associated with the Direct Product of Alternating and Dihedral Groups Acting on the Cartesian Product of Two Sets},
  journal = {American Journal of Applied Mathematics},
  volume = {12},
  number = {6},
  pages = {258-265},
  doi = {10.11648/j.ajam.20241206.15},
  url = {https://doi.org/10.11648/j.ajam.20241206.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241206.15},
  abstract = {Group theory, an area in mathematics, has undergone extensive research, but not exhaustively. In group theory, especially symmetric groups have been studied in terms of properties and actions. In this research, two groups of the Symmetric Group (Alternating and Dihedral Groups) are studied in terms of their properties and also their direct product between them. Also, ordered sets are studied in this research where their Cartesian product is looked at. Finally, this research combines the direct product of two symmetric groups (alternating group and dihedral group) and the Cartesian product of two ordered sets (X×Y) by group action. This research therefore focuses on determining the combinatorial properties (transitivity and primitivity), invariants (ranks and subdegrees), and structures (suborbital graphs) of this group action. To accomplish this, orbit-stabilizer theorem is used to compute transitivity, block action concept is applied to determine primitivity, and Cauchy-Frobenius lemma is used to compute ranks and subdegrees. For n ≥ 3, it has been confirmed that the group action is transitive and also imprimitive. The rank of this group action is 6 and the subdegrees are obtained using the formula of Theorem 3.3. This research adds new concepts in group theory which will be useful in other areas like graph theory and also application in real life situations.},
 year = {2024}
}

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TY  - JOUR
T1  - Combinatorial Properties and Invariants Associated with the Direct Product of Alternating and Dihedral Groups Acting on the Cartesian Product of Two Sets
AU  - John Mokaya Victor
AU  - Nyaga Lewis Namu
AU  - Gikunju David Muriuki
Y1  - 2024/12/18
PY  - 2024
N1  - https://doi.org/10.11648/j.ajam.20241206.15
DO  - 10.11648/j.ajam.20241206.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 258
EP  - 265
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20241206.15
AB  - Group theory, an area in mathematics, has undergone extensive research, but not exhaustively. In group theory, especially symmetric groups have been studied in terms of properties and actions. In this research, two groups of the Symmetric Group (Alternating and Dihedral Groups) are studied in terms of their properties and also their direct product between them. Also, ordered sets are studied in this research where their Cartesian product is looked at. Finally, this research combines the direct product of two symmetric groups (alternating group and dihedral group) and the Cartesian product of two ordered sets (X×Y) by group action. This research therefore focuses on determining the combinatorial properties (transitivity and primitivity), invariants (ranks and subdegrees), and structures (suborbital graphs) of this group action. To accomplish this, orbit-stabilizer theorem is used to compute transitivity, block action concept is applied to determine primitivity, and Cauchy-Frobenius lemma is used to compute ranks and subdegrees. For n ≥ 3, it has been confirmed that the group action is transitive and also imprimitive. The rank of this group action is 6 and the subdegrees are obtained using the formula of Theorem 3.3. This research adds new concepts in group theory which will be useful in other areas like graph theory and also application in real life situations.
VL  - 12
IS  - 6
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Author Information

John Mokaya Victor

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

Contact Email

http://orcid.org/0009-0000-5457-6011
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

Victor, J. M., Namu, N. L., Muriuki, G. D. (2024). Combinatorial Properties and Invariants Associated with the Direct Product of Alternating and Dihedral Groups Acting on the Cartesian Product of Two Sets. American Journal of Applied Mathematics, 12(6), 258-265. https://doi.org/10.11648/j.ajam.20241206.15

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ACS Style

Victor, J. M.; Namu, N. L.; Muriuki, G. D. Combinatorial Properties and Invariants Associated with the Direct Product of Alternating and Dihedral Groups Acting on the Cartesian Product of Two Sets. Am. J. Appl. Math. 2024, 12(6), 258-265. doi: 10.11648/j.ajam.20241206.15

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AMA Style

Victor JM, Namu NL, Muriuki GD. Combinatorial Properties and Invariants Associated with the Direct Product of Alternating and Dihedral Groups Acting on the Cartesian Product of Two Sets. Am J Appl Math. 2024;12(6):258-265. doi: 10.11648/j.ajam.20241206.15

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@article{10.11648/j.ajam.20241206.15,
  author = {John Mokaya Victor and Nyaga Lewis Namu and Gikunju David Muriuki},
  title = {Combinatorial Properties and Invariants Associated with the Direct Product of Alternating and Dihedral Groups Acting on the Cartesian Product of Two Sets},
  journal = {American Journal of Applied Mathematics},
  volume = {12},
  number = {6},
  pages = {258-265},
  doi = {10.11648/j.ajam.20241206.15},
  url = {https://doi.org/10.11648/j.ajam.20241206.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241206.15},
  abstract = {Group theory, an area in mathematics, has undergone extensive research, but not exhaustively. In group theory, especially symmetric groups have been studied in terms of properties and actions. In this research, two groups of the Symmetric Group (Alternating and Dihedral Groups) are studied in terms of their properties and also their direct product between them. Also, ordered sets are studied in this research where their Cartesian product is looked at. Finally, this research combines the direct product of two symmetric groups (alternating group and dihedral group) and the Cartesian product of two ordered sets (X×Y) by group action. This research therefore focuses on determining the combinatorial properties (transitivity and primitivity), invariants (ranks and subdegrees), and structures (suborbital graphs) of this group action. To accomplish this, orbit-stabilizer theorem is used to compute transitivity, block action concept is applied to determine primitivity, and Cauchy-Frobenius lemma is used to compute ranks and subdegrees. For n ≥ 3, it has been confirmed that the group action is transitive and also imprimitive. The rank of this group action is 6 and the subdegrees are obtained using the formula of Theorem 3.3. This research adds new concepts in group theory which will be useful in other areas like graph theory and also application in real life situations.},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - Combinatorial Properties and Invariants Associated with the Direct Product of Alternating and Dihedral Groups Acting on the Cartesian Product of Two Sets
AU  - John Mokaya Victor
AU  - Nyaga Lewis Namu
AU  - Gikunju David Muriuki
Y1  - 2024/12/18
PY  - 2024
N1  - https://doi.org/10.11648/j.ajam.20241206.15
DO  - 10.11648/j.ajam.20241206.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 258
EP  - 265
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20241206.15
AB  - Group theory, an area in mathematics, has undergone extensive research, but not exhaustively. In group theory, especially symmetric groups have been studied in terms of properties and actions. In this research, two groups of the Symmetric Group (Alternating and Dihedral Groups) are studied in terms of their properties and also their direct product between them. Also, ordered sets are studied in this research where their Cartesian product is looked at. Finally, this research combines the direct product of two symmetric groups (alternating group and dihedral group) and the Cartesian product of two ordered sets (X×Y) by group action. This research therefore focuses on determining the combinatorial properties (transitivity and primitivity), invariants (ranks and subdegrees), and structures (suborbital graphs) of this group action. To accomplish this, orbit-stabilizer theorem is used to compute transitivity, block action concept is applied to determine primitivity, and Cauchy-Frobenius lemma is used to compute ranks and subdegrees. For n ≥ 3, it has been confirmed that the group action is transitive and also imprimitive. The rank of this group action is 6 and the subdegrees are obtained using the formula of Theorem 3.3. This research adds new concepts in group theory which will be useful in other areas like graph theory and also application in real life situations.
VL  - 12
IS  - 6
ER  -

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