Which of the following is isomorphic to Z\/5Z ?

a. $ U(5) $

b. $ \mathbb{Z}_5 $

c. $ D_3 $

d. $ S_5 $

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.

Which of the following is isomorphic to Z/5Z?

a. U(5)

b. Z_5

c. D_3

d. S_5

Explore which of the groups U(5), Z_5, D_3, or S_5 is isomorphic to Z/5Z. This problem covers key group theory concepts, including cyclic groups, group isomorphism, and group order. Learn the step-by-step solution, suitable for undergraduate mathematics students.

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