Classification Formula and Generation Algorithm of Cycle Decomposition Expression for Dihedral Groups

Abstract

The necessary of classification research on common formula of $D_n$ group (dihedral group) cycle decomposition expression is illustrated. It includes the reflection and rotation conversion, which derived six common formulae on cycle decomposition expressions of $D_n$ group; it designed the generation algorithm on the cycle decomposition expressions of $D_n$ group, which is based on the method of replacement conversion and the classification formula; algorithm analysis and the results of the process show that the generation algorithm which is based on the classification formula is outperformed by the general algorithm which is based on replacement conversion; it has great significance to solve the enumeration of the necklace combinational scheme, especially the structural problems of combinational scheme, by using group theory and computer.