The Theorem of Cayley and Γ  Matrices

Xiao-Yan Gu; Jian-Qiang Sun

doi:doi:10.11648/j.dmath.20160101.13

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The Theorem of Cayley and Γ Matrices

Xiao-Yan Gu, Jian-Qiang Sun

Published in International Journal of Discrete Mathematics (Volume 1, Issue 1)

Received: 31 October 2016 Accepted: 17 November 2016 Published: 27 December 2016

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Abstract

In this article, the connections between symmetric groups and the matrix groups are investigated for exploring the application of Cayley’s theorem in finite group theory. The exact forms of the permutation groups isomorphic to the groups , and are obtained within the frame of the group-theoretical approach. The results are analyzed in detail and compared with that from Cayley's theorem. It shows that the orders of the symmetric groups in present formulas are less than the latter. Various directions for future investigations on the research results have been pointed out.

Published in	International Journal of Discrete Mathematics (Volume 1, Issue 1)
DOI	10.11648/j.dmath.20160101.13
Page(s)	15-19
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, Isomorphic, γ Matrices, Cayley's Theorem, Quaternion Group

References

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[2]	Nummela, E. C. (1980). Cayley’s Theorem for Topological Groups. Am. Math. Mon., 87 (3), 202-203.
[3]	Crillya, T., Weintraubb, S. H., and Wolfsonc, P. R., (2016). Arthur Cayley, Robert Harley and the quintic equation: newly discovered letters 1859–1863. Historia Mathematica, in press.
[4]	Suksumran, T., Wiboonton, K. (2015). Isomorphism Theorems for Gyrogroups and L-Subgyrogroups. J. Geom. Symmetry Phys., 37, 67-83.
[5]	Childs, L. N., Corradino, J. (2007). Cayley's Theorem and Hopf Galois structures for semidirect products of cyclic groups. J. Algebra, 308 (1), 236-251.
[6]	Roman, S. (2012). Fundamentals of group theory: an advanced approach. New York: Birkhäuser.
[7]	Hamermesh, M. (1962). Group theory and its application to physical problems. Reading: Addison-Wesley.
[8]	Dirac, P. A. M. (1958). The Principles of Quantum Mechanics. London: Oxford University Press.
[9]	Gel'fand, I. M., Minlos, R. A. and Shapiro, Z. Ya. (1963). Representations of the rotation and Lorentz groups and their applications. New York: Pergamon Press.
[10]	Hsu, J.-P. and Zhang, Y.-Z. (2001). Lorentz and Poincare invariance. Singapore: World Scientific.
[11]	Duval, C., Elbistan, M., Horváthy, P. A., Zhang, P.-M. (2015). Wigner–Souriau translations and Lorentz symmetry of chiral fermions. Phys. Lett. B, 742, 322-326.
[12]	Ma, Z.-Q. (2007). Group Theory For Physicists. Singapore: World Scientific.

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Xiao-Yan Gu, Jian-Qiang Sun. (2016). The Theorem of Cayley and Γ Matrices. International Journal of Discrete Mathematics, 1(1), 15-19. https://doi.org/10.11648/j.dmath.20160101.13

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ACS Style

Xiao-Yan Gu; Jian-Qiang Sun. The Theorem of Cayley and Γ Matrices. Int. J. Discrete Math. 2016, 1(1), 15-19. doi: 10.11648/j.dmath.20160101.13

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AMA Style

Xiao-Yan Gu, Jian-Qiang Sun. The Theorem of Cayley and Γ Matrices. Int J Discrete Math. 2016;1(1):15-19. doi: 10.11648/j.dmath.20160101.13

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@article{10.11648/j.dmath.20160101.13,
  author = {Xiao-Yan Gu and Jian-Qiang Sun},
  title = {The Theorem of Cayley and Γ  Matrices},
  journal = {International Journal of Discrete Mathematics},
  volume = {1},
  number = {1},
  pages = {15-19},
  doi = {10.11648/j.dmath.20160101.13},
  url = {https://doi.org/10.11648/j.dmath.20160101.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20160101.13},
  abstract = {In this article, the connections between symmetric groups and the matrix groups  are investigated for exploring the application of Cayley’s theorem in finite group theory. The exact forms of the permutation groups isomorphic to the groups ,  and  are obtained within the frame of the group-theoretical approach. The results are analyzed in detail and compared with that from Cayley's theorem. It shows that the orders of the symmetric groups in present formulas are less than the latter. Various directions for future investigations on the research results have been pointed out.},
 year = {2016}
}

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TY  - JOUR
T1  - The Theorem of Cayley and Γ  Matrices
AU  - Xiao-Yan Gu
AU  - Jian-Qiang Sun
Y1  - 2016/12/27
PY  - 2016
N1  - https://doi.org/10.11648/j.dmath.20160101.13
DO  - 10.11648/j.dmath.20160101.13
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 15
EP  - 19
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20160101.13
AB  - In this article, the connections between symmetric groups and the matrix groups  are investigated for exploring the application of Cayley’s theorem in finite group theory. The exact forms of the permutation groups isomorphic to the groups ,  and  are obtained within the frame of the group-theoretical approach. The results are analyzed in detail and compared with that from Cayley's theorem. It shows that the orders of the symmetric groups in present formulas are less than the latter. Various directions for future investigations on the research results have been pointed out.
VL  - 1
IS  - 1
ER  -

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Author Information

Xiao-Yan Gu

Department of Physics, East China University of Science and Technology, Shanghai, China
Jian-Qiang Sun

College of Information Science and Technology, Hainan University, Haikou, China

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Xiao-Yan Gu, Jian-Qiang Sun. (2016). The Theorem of Cayley and Γ Matrices. International Journal of Discrete Mathematics, 1(1), 15-19. https://doi.org/10.11648/j.dmath.20160101.13

Copy | Download

ACS Style

Xiao-Yan Gu; Jian-Qiang Sun. The Theorem of Cayley and Γ Matrices. Int. J. Discrete Math. 2016, 1(1), 15-19. doi: 10.11648/j.dmath.20160101.13

Copy | Download

AMA Style

Xiao-Yan Gu, Jian-Qiang Sun. The Theorem of Cayley and Γ Matrices. Int J Discrete Math. 2016;1(1):15-19. doi: 10.11648/j.dmath.20160101.13

Copy | Download

@article{10.11648/j.dmath.20160101.13,
  author = {Xiao-Yan Gu and Jian-Qiang Sun},
  title = {The Theorem of Cayley and Γ  Matrices},
  journal = {International Journal of Discrete Mathematics},
  volume = {1},
  number = {1},
  pages = {15-19},
  doi = {10.11648/j.dmath.20160101.13},
  url = {https://doi.org/10.11648/j.dmath.20160101.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20160101.13},
  abstract = {In this article, the connections between symmetric groups and the matrix groups  are investigated for exploring the application of Cayley’s theorem in finite group theory. The exact forms of the permutation groups isomorphic to the groups ,  and  are obtained within the frame of the group-theoretical approach. The results are analyzed in detail and compared with that from Cayley's theorem. It shows that the orders of the symmetric groups in present formulas are less than the latter. Various directions for future investigations on the research results have been pointed out.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - The Theorem of Cayley and Γ  Matrices
AU  - Xiao-Yan Gu
AU  - Jian-Qiang Sun
Y1  - 2016/12/27
PY  - 2016
N1  - https://doi.org/10.11648/j.dmath.20160101.13
DO  - 10.11648/j.dmath.20160101.13
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 15
EP  - 19
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20160101.13
AB  - In this article, the connections between symmetric groups and the matrix groups  are investigated for exploring the application of Cayley’s theorem in finite group theory. The exact forms of the permutation groups isomorphic to the groups ,  and  are obtained within the frame of the group-theoretical approach. The results are analyzed in detail and compared with that from Cayley's theorem. It shows that the orders of the symmetric groups in present formulas are less than the latter. Various directions for future investigations on the research results have been pointed out.
VL  - 1
IS  - 1
ER  -

Copy | Download