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We introduce several ways of producing subgroups of U(n), the group of units of  Z n. 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.

Factoring Subgroups and Factor Groups of the Unit Group Modulo n