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Factoring Subgroups and Factor Groups of the Unit Group Modulo n

Summary We introduce several ways of producing subgroups of U(n), the group of units of Zn . Our results find the structure of these various subgroups in terms of external direct product of cyclic groups. We then use our classifications to give descriptions of elements of U(n) that form a subgroup of U(n) with a desired cyclic group decomposition. This includes a description of elements that form a Sylow p-subgroup.