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Browsing by Author "Bagalá, Nicola"

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  • Bagalá, Nicola (2017)
    Given two groups, there are several ways of obtaining news ones. This work focuses on three of these ways: the direct, semidirect, and wreath products. These three products can be thought of as subsequently 'building upon' each other, since the definition of semidirect product depends on the concept of direct product, and wreath products are essentially a particular example of semidirect product. The concepts above were explored both theoretically and practically, by means of several different examples as well as some digressions from the main topics for the benefit of interested readers. The most substantial and convoluted examples of semidirect and wreath products were given in the last section, where the algebraic structures of Rubik's group and of the illegal Rubik's group are introduced. These are the groups of, respectively, all legal and possible (legal or illegal) moves one can perform on Rubik's cube. An illegal move is such that it cannot be performed without taking the cube apart and reassembling it differently. Rubik's group is generated by all legal basic moves that can be performed on Rubik's cube - for example, twisting a face of the cube left or right. This extremely large-sized group contains two particular subgroups, namely the subgroups of orientation-preserving and position-preserving moves. The first is such that any of the moves in it, if applied to the cube, will leave the orientation of all the cube's 'cubies' unchanged, with respect to a labelling system priorly established on the cube itself, though they may change the position of the cubies. Similarly, the elements of the subgroup of position-preserving moves will not change the position of the cubies, but they may change their orientation. The main result proved in this work is that the legal Rubik's group is the semidirect product of the orientation-preserving and position-preserving subgroups. The method used is mainly based on, and it expands upon, that used by Charles Bandelow in his book Inside Rubik's cube and beyond. A second fact - that the illegal Rubik's group is isomorphic to a direct product of wreath products - was also proved as a secondary goal.