ON THE PRIMITIVE AND REGULAR CHARACTERISTICS OF DIHEDRAL GROUP OF DEGREE 2P AND THEIR RELEVANCE TO MUSICAL NOTE THEORY
Abstract
This paper investigates the primitive and regular characteristics of Dihedral Group of degree 2p, where p is an odd prime. By utilizing numerical approached, the properties of these groups were examined to shed light on their structure, behavior, and underlying algebraic characteristics. The work uses some group concept to test conditions for primitivity and regularity in these groups, with the help of Group Algorithm Programming (GAP) our results were validated. The main focus of this paper is on their applications to musical note theory. We explore the conditions under which these groups exhibit primitive and regular action on sets, highlighting their algebraic properties and symmetries. The theoretical findings are then connected to musical note arrangements, where pitch classes and transformations exhibit similar cyclic and reflective patterns. By establishing this connection, we demonstrate how group-theoretic principles can enhance the understanding of musical scales, chord structures, and symmetrical note sequences. The results presented offer new insights into the intersection of abstract algebra and music, paving the way for further interdisciplinary exploration. The work reveal that the musical note operate base on their pitch classes and musical intervals. It was discovered that the group of transpositions and inversions, denoted T_nT_(1-n) is isomorphic to the dihedral group 12. Finally, Its Conjugacy classes and Character table was presented.
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