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This project concerns with a special type of subgroups of the group of rational modulo one. The elements in this subgroup G are the congruence class of rational numbers whose denominators are a power of a prime number. Specifically, the project will investigate the following problems:

1. The number of elements of G.

2. The generators of G.

3. The structures of the subgroups of G.

4. The order of every element in this group.

5. Estimation of the upper bound of the orders of all the elements in G.