U(30) refers to the set of integers that are relatively prime to 30, and the operation used is multiplication modulo 30. This set forms a group under this operation, and it is known as the group of units modulo 30.
Cyclic subgroups are subgroups of a group that are generated by a single element. In other words, every element in a cyclic subgroup can be written as a power of a specific element, known as the generator.
To find the cyclic subgroups of U(30), you can start by listing out all the elements in U(30). Then, choose an element as the generator and find all the possible powers of that element. The set of all these powers will form a cyclic subgroup of U(30).
The order of U(30) is the number of elements in this group. In this case, U(30) has 8 elements, which are 1, 7, 11, 13, 17, 19, 23, and 29. This means that there are 8 possible cyclic subgroups of U(30).
One possible cyclic subgroup of U(30) is the subgroup generated by the element 7. The powers of 7 in U(30) are 7, 13, 19, 23, 29, 11, 17, and 1. Therefore, this subgroup is {1, 7, 11, 13, 17, 19, 23, 29}.