The group that is isomorphic to $\mathbb{Z}/5\mathbb{Z}$ (also denoted as $\mathbb{Z}_5$) is the one that shares the same structure, meaning it must be cyclic of order 5.

Let's go through the options:

• a. $U(5)$: This is the group of units (invertible elements) in $\mathbb{Z}/5\mathbb{Z}$. For $n = 5$, $U(5)$ consists of the elements $\{1, 2, 3, 4\}$ under multiplication modulo 5. There are 4 elements, so the order of $U(5)$ is 4, not 5. Hence, it is not isomorphic to $\mathbb{Z}/5\mathbb{Z}$.

• b. $\mathbb{Z}_5$: This is another notation for $\mathbb{Z}/5\mathbb{Z}$, the cyclic group of order 5. So, it is clearly isomorphic to itself. Therefore, this is the correct answer.

• c. $D_3$: This is the dihedral group of order 6, which represents the symmetries of an equilateral triangle. Since its order is 6, it cannot be isomorphic to $\mathbb{Z}/5\mathbb{Z}$.

• d. $S_5$: This is the symmetric group on 5 elements, which has an order of 120. Since the order is much larger than 5, it is not isomorphic to $\mathbb{Z}/5\mathbb{Z}$.

Thus, the correct answer is b. $\mathbb{Z}_5$.

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Would you like more details or have any further questions?

Follow-up Questions:

1. What is the general structure of cyclic groups?
2. Can you explain the group $U(n)$ for other values of $n$?
3. How is the order of a group related to its isomorphism class?
4. What is the structure of $D_n$ for general $n$?
5. What properties of symmetric groups make them non-isomorphic to cyclic groups?

Tip:

When checking if two groups are isomorphic, compare their structure, such as order, cyclic nature, and types of elements.