1. **Problem Statement:** Consider the group $\mathbb{Z}_5 = \{0,1,2,3,4\}$ under addition modulo 5. Find the order of $\mathbb{Z}_5$ and determine whether each element is a generator of the group. 2. **Order of the group:** The order of a group is the number of elements in it. Here, $\mathbb{Z}_5$ has elements $0,1,2,3,4$, so its order is: $$|\mathbb{Z}_5| = 5$$ 3. **Generators of $\mathbb{Z}_5$:** A generator $g$ of a cyclic group $G$ is an element such that every element of $G$ can be written as $g^k$ for some integer $k$. For additive groups like $\mathbb{Z}_5$, this means the set $\{g, 2g, 3g, \ldots\}$ modulo 5 covers all elements. 4. **Check each element:** - For $0$: $\{0, 0+0, 0+0+0, \ldots\} = \{0\}$ only, so not a generator. - For $1$: $\{1, 2, 3, 4, 0\}$ modulo 5 covers all elements, so $1$ is a generator. - For $2$: $\{2, 4, 1, 3, 0\}$ modulo 5 covers all elements, so $2$ is a generator. - For $3$: $\{3, 1, 4, 2, 0\}$ modulo 5 covers all elements, so $3$ is a generator. - For $4$: $\{4, 3, 2, 1, 0\}$ modulo 5 covers all elements, so $4$ is a generator. 5. **Summary:** The order of $\mathbb{Z}_5$ is $5$. All elements except $0$ are generators of $\mathbb{Z}_5$. **Final answer:** - Order of $\mathbb{Z}_5$ is $5$. - Generators are $1, 2, 3, 4$.