Structures 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.20241206.17

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Structures 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: 1 December 2024 Accepted: 12 December 2024 Published: 30 December 2024

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Abstract

Whenever a permutation group acts on a set, combinatorial and invariant properties, and mathematical structures that result from this group action are studied. Various mathematicians have studied these properties over time using different groups acting on both ordered and unordered sets. The combinatorial properties (transitivity and primitivity) and invariants (ranks and subdegrees) of the direct product between alternating and dihedral groups acting on the Cartesian product of two sets have alreadybeenstudiedanditwasfoundoutthatthegroupactionistransitive, imprimitive, therankis6, andsubdegreesareobtained according to theorem 2.3. This research seeks to extend this by constructing and analyzing the properties (simple/multigraph, self-pairedness, connectedness, degree of the vertex, girth, and directedness) of these mathematical structures (suborbital graphs) that result from the group action. This research for n ≥ 3, suborbital graphs can be classified into three categories; First, those constructed when only the first components of the vertex set are identical and second, those when only the second components of the vertex set are identical. The suborbital graphs of the first and second category are simple, self-paired, have n− disconnected components, are regular with degree n − 1 and girth is 3. The third category of suborbital graphs in which neither the first nor the second components of the vertex set are identical and they are; simple, self-paired, connected, regular with degree of vertex varying from graph to graph, and girth 3.

Published in	American Journal of Applied Mathematics (Volume 12, Issue 6)
DOI	10.11648/j.ajam.20241206.17
Page(s)	286-292
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

Permutation, Group Action, Alternating Group, Dihedral Group, Suborbital, Suborbital Graph

References

[1]	Shanker, G. R. (2009). Discrete Mathematical Structures. New Age International (p) Limited Publishers, 2, 218.
[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- 387-21768-1 4
[3]	Burnside William. (1911). Theory of Groups of finite Order. Cambridge University Press. 2. 1.
[4]	Thomas, W. J. (2010). Abstract Algebra (Theory and Applications). Stephen F. Austin University.
[5]	John, B. F. (2003). A First Course in Abstract Algebra. Pearson Education Inc. 7th Edition.
[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]	Nyaga, L. N., Kamuti, I. N., Mwathi, C. W., Akanga, J. R. (2011). Ranks and Subdegrees of the Symmetric Group Sn Acting on Unordered r-element Subsets. http://ir.jkuat.ac.ke/handle/123456789/918
[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]	Edgar, G. G., Michael, M. P. (2002). Discrete Mathematics with Group Theory. Prentice-Hall Inc, 2, 286.
[11]	Douglas, E. E., Winston, J. C. (2006). Discrete Mathematics. John Wiley and Sons Inc, 1, 510.
[12]	Seymour Lipschutz, Marc Lars Lipson. (2007). Discrete Mathematics. Mc Graw-Hill companies, 3, 157.
[13]	Oscar Levin. (2021). Discrete Mathematics. 3, 243. https://discrete.openmathbooks.org/dmoi3.html
[14]	Richard Johnsonbaugh. (2018). Discrete Mathematics. Pearson, 8th Edition, 385.
[15]	Liu, C. L.(2001). Elements of Discrete Mathematics. Tata McGraw-Hill Publishing Company, 2, 139.
[16]	Richard, A. Gibbs. (1974). Self-Complementary Groups. Journal of Combinatorial Theory (B), 16, 106- 123. https://doi.org/10.1016/0095-8956(74)90053-7
[17]	Helmut Wielandth. (1964). Multiply Transitive Groups. Finite Permutation Groups, 19-43. https://doi.org/10.1016/B978-0-12-749650-4.50007-2
[18]	Cameron, P. J. (1975). Suborbits in transitive permutation groups. Proceedings of the NATO Advanced Study Institute on Combinatorics, 16: 419- 450. https://doi.org/10.1007/978-94-010-1826-5 20
[19]	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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APA Style

Victor, J. M., Namu, N. L., Muriuki, G. D. (2024). Structures 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), 286-292. https://doi.org/10.11648/j.ajam.20241206.17

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ACS Style

Victor, J. M.; Namu, N. L.; Muriuki, G. D. Structures 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), 286-292. doi: 10.11648/j.ajam.20241206.17

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AMA Style

Victor JM, Namu NL, Muriuki GD. Structures 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):286-292. doi: 10.11648/j.ajam.20241206.17

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@article{10.11648/j.ajam.20241206.17,
  author = {John Mokaya Victor and Nyaga Lewis Namu and Gikunju David Muriuki},
  title = {Structures 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 = {286-292},
  doi = {10.11648/j.ajam.20241206.17},
  url = {https://doi.org/10.11648/j.ajam.20241206.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241206.17},
  abstract = {Whenever a permutation group acts on a set, combinatorial and invariant properties, and mathematical structures that result from this group action are studied. Various mathematicians have studied these properties over time using different groups acting on both ordered and unordered sets. The combinatorial properties (transitivity and primitivity) and invariants (ranks and subdegrees) of the direct product between alternating and dihedral groups acting on the Cartesian product of two sets have alreadybeenstudiedanditwasfoundoutthatthegroupactionistransitive, imprimitive, therankis6, andsubdegreesareobtained according to theorem 2.3. This research seeks to extend this by constructing and analyzing the properties (simple/multigraph, self-pairedness, connectedness, degree of the vertex, girth, and directedness) of these mathematical structures (suborbital graphs) that result from the group action. This research for n ≥ 3, suborbital graphs can be classified into three categories; First, those constructed when only the first components of the vertex set are identical and second, those when only the second components of the vertex set are identical. The suborbital graphs of the first and second category are simple, self-paired, have n− disconnected components, are regular with degree n − 1 and girth is 3. The third category of suborbital graphs in which neither the first nor the second components of the vertex set are identical and they are; simple, self-paired, connected, regular with degree of vertex varying from graph to graph, and girth 3.},
 year = {2024}
}

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TY  - JOUR
T1  - Structures 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  - 2024/12/30
PY  - 2024
N1  - https://doi.org/10.11648/j.ajam.20241206.17
DO  - 10.11648/j.ajam.20241206.17
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 286
EP  - 292
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20241206.17
AB  - Whenever a permutation group acts on a set, combinatorial and invariant properties, and mathematical structures that result from this group action are studied. Various mathematicians have studied these properties over time using different groups acting on both ordered and unordered sets. The combinatorial properties (transitivity and primitivity) and invariants (ranks and subdegrees) of the direct product between alternating and dihedral groups acting on the Cartesian product of two sets have alreadybeenstudiedanditwasfoundoutthatthegroupactionistransitive, imprimitive, therankis6, andsubdegreesareobtained according to theorem 2.3. This research seeks to extend this by constructing and analyzing the properties (simple/multigraph, self-pairedness, connectedness, degree of the vertex, girth, and directedness) of these mathematical structures (suborbital graphs) that result from the group action. This research for n ≥ 3, suborbital graphs can be classified into three categories; First, those constructed when only the first components of the vertex set are identical and second, those when only the second components of the vertex set are identical. The suborbital graphs of the first and second category are simple, self-paired, have n− disconnected components, are regular with degree n − 1 and girth is 3. The third category of suborbital graphs in which neither the first nor the second components of the vertex set are identical and they are; simple, self-paired, connected, regular with degree of vertex varying from graph to graph, and girth 3.
VL  - 12
IS  - 6
ER  -

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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). Structures 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), 286-292. https://doi.org/10.11648/j.ajam.20241206.17

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ACS Style

Victor, J. M.; Namu, N. L.; Muriuki, G. D. Structures 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), 286-292. doi: 10.11648/j.ajam.20241206.17

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AMA Style

Victor JM, Namu NL, Muriuki GD. Structures 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):286-292. doi: 10.11648/j.ajam.20241206.17

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@article{10.11648/j.ajam.20241206.17,
  author = {John Mokaya Victor and Nyaga Lewis Namu and Gikunju David Muriuki},
  title = {Structures 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 = {286-292},
  doi = {10.11648/j.ajam.20241206.17},
  url = {https://doi.org/10.11648/j.ajam.20241206.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241206.17},
  abstract = {Whenever a permutation group acts on a set, combinatorial and invariant properties, and mathematical structures that result from this group action are studied. Various mathematicians have studied these properties over time using different groups acting on both ordered and unordered sets. The combinatorial properties (transitivity and primitivity) and invariants (ranks and subdegrees) of the direct product between alternating and dihedral groups acting on the Cartesian product of two sets have alreadybeenstudiedanditwasfoundoutthatthegroupactionistransitive, imprimitive, therankis6, andsubdegreesareobtained according to theorem 2.3. This research seeks to extend this by constructing and analyzing the properties (simple/multigraph, self-pairedness, connectedness, degree of the vertex, girth, and directedness) of these mathematical structures (suborbital graphs) that result from the group action. This research for n ≥ 3, suborbital graphs can be classified into three categories; First, those constructed when only the first components of the vertex set are identical and second, those when only the second components of the vertex set are identical. The suborbital graphs of the first and second category are simple, self-paired, have n− disconnected components, are regular with degree n − 1 and girth is 3. The third category of suborbital graphs in which neither the first nor the second components of the vertex set are identical and they are; simple, self-paired, connected, regular with degree of vertex varying from graph to graph, and girth 3.},
 year = {2024}
}

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TY  - JOUR
T1  - Structures 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  - 2024/12/30
PY  - 2024
N1  - https://doi.org/10.11648/j.ajam.20241206.17
DO  - 10.11648/j.ajam.20241206.17
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 286
EP  - 292
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20241206.17
AB  - Whenever a permutation group acts on a set, combinatorial and invariant properties, and mathematical structures that result from this group action are studied. Various mathematicians have studied these properties over time using different groups acting on both ordered and unordered sets. The combinatorial properties (transitivity and primitivity) and invariants (ranks and subdegrees) of the direct product between alternating and dihedral groups acting on the Cartesian product of two sets have alreadybeenstudiedanditwasfoundoutthatthegroupactionistransitive, imprimitive, therankis6, andsubdegreesareobtained according to theorem 2.3. This research seeks to extend this by constructing and analyzing the properties (simple/multigraph, self-pairedness, connectedness, degree of the vertex, girth, and directedness) of these mathematical structures (suborbital graphs) that result from the group action. This research for n ≥ 3, suborbital graphs can be classified into three categories; First, those constructed when only the first components of the vertex set are identical and second, those when only the second components of the vertex set are identical. The suborbital graphs of the first and second category are simple, self-paired, have n− disconnected components, are regular with degree n − 1 and girth is 3. The third category of suborbital graphs in which neither the first nor the second components of the vertex set are identical and they are; simple, self-paired, connected, regular with degree of vertex varying from graph to graph, and girth 3.
VL  - 12
IS  - 6
ER  -

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