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

a. $ U(5) $

b. $ \mathbb{Z}_5 $

c. $ D_3 $

d. $ S_5 $

The correct answer is **b. $ \mathbb{Z}_5 $**.

### Explanation:
1. $ \mathbb{Z}/5\mathbb{Z} $ is the group of integers modulo 5, also denoted as $ \mathbb{Z}_5 $. It consists of the elements $ \{ 0, 1, 2, 3, 4 \} $ under addition modulo 5, and it forms a cyclic group of order 5.
   
   Now, let's review the other options:
   
- **a. $ U(5) $:**  
  The group $ U(5) $ refers to the group of units (invertible elements) modulo 5, which are $ \{ 1, 2, 3, 4 \} $. Since $ U(5) $ has only 4 elements, it cannot be isomorphic to $ \mathbb{Z}/5\mathbb{Z} $, which has 5 elements.

- **b. $ \mathbb{Z}_5 $:**  
  This is simply another notation for $ \mathbb{Z}/5\mathbb{Z} $, so this group is clearly isomorphic to itself.

- **c. $ D_3 $ (Dihedral group of order 6):**  
  $ D_3 $ is the symmetry group of an equilateral triangle, with 6 elements. Since it has more elements than $ \mathbb{Z}/5\mathbb{Z} $, they are not isomorphic.

- **d. $ S_5 $ (Symmetric group on 5 elements):**  
  $ S_5 $ is the group of all permutations on 5 elements, with 120 elements. Again, since this group has more elements than $ \mathbb{Z}/5\mathbb{Z} $, they are not isomorphic.

Thus, the only isomorphic group is $ \mathbb{Z}_5 $.

### Follow-up Questions:
1. What are the generators of the group $ \mathbb{Z}_5 $?
2. Can $ U(5) $ be isomorphic to any other cyclic group?
3. What are the orders of the elements in $ D_3 $?
4. How many elements does $ S_5 $ have, and what is its structure?
5. Can a dihedral group ever be isomorphic to a cyclic group?

### Tip:
When comparing groups for isomorphisms, first check if they have the same number of elements (order) as a basic prerequisite.

Which of the following is isomorphic to Z/5Z?
a. U(5)
b. Z_5
c. D_3
d. S_5

This page covers the identification of which group is isomorphic to Z/5Z, focusing on concepts from group theory such as cyclic groups, group order, and isomorphisms. The correct solution is Z_5, and detailed explanations are provided for why the other options (U(5), D_3, S_5) are not isomorphic. Suitable for undergraduate abstract algebra students.

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