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

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21. A cyclic group of order 1 has only one element, which is the identity element. The only generator of a cyclic group is the identity itself. Therefore, a cyclic group of order 1 has **1 generator**.

22. To find the generators of the cyclic groups $$ \mathbb{Z}_{12} $$, $$ \mathbb{Z}_{15} $$, and $$ \mathbb{Z}_{23} $$, we need to identify the elements that are coprime to the order of the group.

- **Generators of $$ \mathbb{Z}_{12} $$**: The elements of $$ \mathbb{Z}_{12} $$ are $$ \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\} $$. The generators are those elements that are coprime to 12. The coprime elements are:
  - $$ 1 $$ (gcd(1, 12) = 1)
  - $$ 5 $$ (gcd(5, 12) = 1)
  - $$ 7 $$ (gcd(7, 12) = 1)
  - $$ 11 $$ (gcd(11, 12) = 1)

Therefore, the generators of $$ \mathbb{Z}_{12} $$ are $$ \{1, 5, 7, 11\} $$.

- **Generators of $$ \mathbb{Z}_{15} $$**: The elements of $$ \mathbb{Z}_{15} $$ are $$ \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\} $$. The coprime elements are:
  - $$ 1 $$ (gcd(1, 15) = 1)
  - $$ 2 $$ (gcd(2, 15) = 1)
  - $$ 4 $$ (gcd(4, 15) = 1)
  - $$ 7 $$ (gcd(7, 15) = 1)
  - $$ 8 $$ (gcd(8, 15) = 1)
  - $$ 11 $$ (gcd(11, 15) = 1)
  - $$ 13 $$ (gcd(13, 15) = 1)
  - $$ 14 $$ (gcd(14, 15) = 1)

Therefore, the generators of $$ \mathbb{Z}_{15} $$ are $$ \{1, 2, 4, 7, 8, 11, 13, 14\} $$.

- **Generators of $$ \mathbb{Z}_{23} $$**: Since 23 is a prime number, all non-zero elements are coprime to 23. The elements of $$ \mathbb{Z}_{23} $$ are $$ \{0, 1, 2, \ldots, 22\} $$. The generators are:
  - $$ 1, 2, 3, \ldots, 22 $$ (all non-zero elements)

Therefore, the generators of $$ \mathbb{Z}_{23} $$ are $$ \{1, 2, 3, \ldots, 22\} $$.

In summary:
- Generators of $$ \mathbb{Z}_{12} $$: $$ \{1, 5, 7, 11\} $$
- Generators of $$ \mathbb{Z}_{15} $$: $$ \{1, 2, 4, 7, 8, 11, 13, 14\} $$
- Generators of $$ \mathbb{Z}_{23} $$: $$ \{1, 2, 3, \ldots, 22\} $$
21. A cyclic group of order 1 has 1 generator. 22. Generators of $$ \mathbb{Z}_{12} $$: $$ \{1, 5, 7, 11\} $$; Generators of $$ \mathbb{Z}_{15} $$: $$ \{1, 2, 4, 7, 8, 11, 13, 14\} $$; Generators of $$ \mathbb{Z}_{23} $$: $$ \{1, 2, 3, \ldots, 22\} $$

How many gencrators can a cyclic group of order 1

Oive the generators of ?

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