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The success of Group Theory is impressive and extraordinary. Its influence is strongly felt in almost all scientific and artistic disciplines, in Music in particular, as is shown in this text. Group Theory extracts the essential characteristics of diverse situations in which some type of symmetry or transformation appears. In this text, a modern presentation of the fundamental notions of Group Theory is chosen, where the language of commutative diagrams and universal properties, so necessary in Modern Mathematics, in Physics and Computer Science, among other disciplines, is introduced.