The purpose of the project is better understand the theory of groups, especially the structure theory of groups, including substructures of groups and the morphisms between groups.
Congruences on a group determine quotient groups. In the additive group of rational numbers is there a congruence defined by: two rational numbers are congruent if and only if their difference is an integer. The quotient group obtained from this congruence is an important commutative group, called the group of rational modulo one.
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.