To determine all the generators of a group of order 17, let's break it down step-by-step.

Key points:

1. A group of order 17 is a cyclic group, since 17 is prime. A cyclic group of prime order has the structure $\mathbb{Z}_{17}$, meaning that every element in the group can be written as powers of a single element (called a generator).

2. In a cyclic group $\mathbb{Z}_n$ of prime order $n$, an element $g$ is a generator if the smallest positive integer $k$ such that $g^k = 1$ is $n$. This means that $g$ generates all the elements $1, 2, 3, \ldots, 16$ under its powers.

3. For a cyclic group of order 17, the generators are the elements $g \in \mathbb{Z}_{17}$ such that $\text{gcd}(g, 17) = 1$, where gcd denotes the greatest common divisor.

Step-by-step process:

1. The group $\mathbb{Z}_{17} = \{0, 1, 2, \ldots, 16\}$.
2. The element 0 does not generate the group, because its powers are always 0.
3. The element $g$ generates the group if and only if $\gcd(g, 17) = 1$.
4. The elements $g$ such that $\gcd(g, 17) = 1$ are the integers from 1 to 16, since 17 is prime and the only divisor of 17 is 1.

Conclusion:

The generators of the group $\mathbb{Z}_{17}$ are the elements $g \in \{1, 2, 3, \ldots, 16\}$ such that $\gcd(g, 17) = 1$. Since 17 is prime, all non-zero elements are coprime to 17. Therefore, the generators are:

$g \in \{1, 2, 3, \ldots, 16\}$

Specifically, the elements 1, 2, 3, ..., 16 are all possible generators of this cyclic group, excluding 0.

Do you have any specific questions on this topic or need further clarification?

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Here are some related questions to expand your understanding:

1. What is the definition of a cyclic group?
2. How can we determine the order of an element in a group?
3. What are the properties of a prime order group?
4. Why are all elements in $\mathbb{Z}_{17}$ (except 0) generators?
5. Can you explain how the greatest common divisor (gcd) helps find generators?

Tip: In a cyclic group of prime order, all non-identity elements are generators!