The Application and Extension of Group Theory in Rubik's Cube
Keywords:
Group Theory, Cude, Permutation GroupsAbstract
The paper aims to explore the practical application of group theory in the analysis of Rubik’s Cube. As a highly challenging and intellectual three-dimensional puzzle, the Rubik’s Cube has attracted widespread research interest. By applying the concepts and techniques of permutation groups in group theory, we further explore the properties of Rubik’s Cube transformations. Firstly, we introduce the basic concepts of group theory and the definition of permutation groups, as well as the background and objectives of the Rubik’s Cube problem. Then, we discuss in detail how to convert Rubik’s Cube operations and rotations into elements and operations of permutation groups, to analyze and solve Rubik’s Cube problems within an abstract algebraic framework. We explain the properties and structure of permutation groups, such as group order, subgroups, and generators, as well as how to use these concepts to simplify the process of solving Rubik’s Cube. We also introduce some classical group theory techniques and algorithms, such as inverse elements, exponentiation, and cycle notation, and explain their practical application in Rubik’s Cube analysis.
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