Introduction of Several Special Groups and Their Applications to Rubik’s Cube
DOI:
https://doi.org/10.54097/hset.v47i.8186Keywords:
Group theory, Permutation group, Cayley’s theorem, Rubik’s cube.Abstract
Group theory is the subject that aims to study the symmetries and structures of groups in mathematics. This work provides an introduction to group theory and explores some potential applications of group theory on complex geometric objects like the Rubik's cube. To this end, the concepts of symmetric group, permutation group, and cyclic group are introduced, and the famous Lagrange’s theorem and Cayley’s theorem are mentioned briefly. The former theorem establishes that a subgroup’s order must be a divisor of the parent group’s order. Concerning the permutation group, it is a set of permutations that form a group under composition. Hence, the various groups that can be formed by the Rubik's cube are discussed, including the group of all possible permutations of the cube's stickers, and the subgroups that are generated through permutations of the six basic movements embedded in Rubik’s cube. Overall, this essay provides an accessible introduction to group theory and its applications to the popular Rubik's cube.
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References
Dummit David, Foote Richard. Abstract algebra. Wiley, Vermont, 2004.
Wang Efang. Fundamentals of finite group theory. Tsinghua University Press, 2002.
Farahat H. K., Peel M. H. On the representation theory of the symmetric groups. Journal of Algebra, 1980, 67(2): 280-304.
Cameron Peter J. and Semeraro Jason. The Cycle Polynomial of a Permutation Group. The Electronic Journal of Combinatorics, 2018, 25(1): 1-14.
Gopalakrishnan Mini, Kumari N. Naga Maruthi. Generator graphs for cyclic groups. AIP Conference Proceedings, 2019, 2112(1): 020119.
Mamidi Sai Akash. Applications Of Lagrange’s Theorem in Group Theory. Int. J. Math. Comput. Sci., 2015, 3(8): 1150-1153.
Graves-Morris P. R., Baker George A., Woodcock C. F. Cayley's theorem and its application in the theory of vector Padé approximants, J. Comput. Appl. Math, 1996, 66(1): 255-265.
Joyner David. Adventures in group theory: Rubik’s cube, Merlin’s machine, and other mathematical toys. John Hopkins University Press, 2008.
Shih K.-S. Weighted Permutations and the Group of the Rubik’s Cube. Chinese Journal of Mathematics, 1981, 9(2): 65–78.
Volte E., Patarin J., Nachef V. Zero knowledge with Rubik’s cubes and non-abelian groups. Cryptology and Network Security, 2013, 8257: 74-91.
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