21. How many gencrators can a cyclic group of order 1 22. Oive the generators of \( z_{12}, z_{15}, z_{23} \) ?

In a cyclic group of order 1, there is only one element, which is the identity element itself. Hence, this single element serves as the only generator of the group since it can produce itself through repeated application of the group operation. For the groups \( \mathbb{Z}_{12} \), \( \mathbb{Z}_{15} \), and \( \mathbb{Z}_{23} \), the generators can be found by identifying the elements that are coprime to the order of the group. For \( \mathbb{Z}_{12} \), the generators are 1, 5, 7, and 11. For \( \mathbb{Z}_{15} \), the generators are 1, 2, 4, 7, 8, 11, 13, and 14. Lastly, for \( \mathbb{Z}_{23} \) (which is prime), all non-zero elements from 1 to 22 are generators. So, there you have it—powerful little communities in every group!