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The problem involves determining the homomorphic images of \( U(8) \), the group of units modulo 8, and relates to the First Isomorphism Theorem.

### Step 1: Structure of \( U(8) \)
The group \( U(8) \) consists of the elements that are relatively prime to 8 under multiplication modulo 8. These elements are:
\[
U(8) = \{1, 3, 5, 7\}
\]
This is a cyclic group of order 4, generated by 3. So, \( U(8) \cong \mathbb{Z}_4 \) (the cyclic group of order 4).

### Step 2: Possible Homomorphic Images
By the First Isomorphism Theorem, the homomorphic images of a group are determined by its quotient groups. Since \( U(8) \cong \mathbb{Z}_4 \), we need to look at all possible quotient groups of \( \mathbb{Z}_4 \). These will be the homomorphic images.

The subgroups of \( \mathbb{Z}_4 \) are:
1. \( \{0\} \) (the trivial group, isomorphic to \( \mathbb{Z}_1 \)),
2. \( \mathbb{Z}_2 \) (the cyclic group of order 2),
3. \( \mathbb{Z}_4 \) itself.

The possible quotient groups are then:
- \( \mathbb{Z}_4 / \mathbb{Z}_4 = \{0\} \cong \mathbb{Z}_1 \),
- \( \mathbb{Z}_4 / \mathbb{Z}_2 \cong \mathbb{Z}_2 \),
- \( \mathbb{Z}_4 / \{0\} \cong \mathbb{Z}_4 \).

Thus, the homomorphic images of \( U(8) \) are:
- \( \mathbb{Z}_4 \),
- \( \mathbb{Z}_2 \),
- \( \mathbb{Z}_1 \) (the trivial group).

### Step 3: Evaluating the Answer Choices
The question asks which of the provided lists contains all homomorphic images of \( U(8) \) up to isomorphism.

- **A: \( U(8), \mathbb{Z}_1 \)** — This list is incomplete because it does not include \( \mathbb{Z}_2 \) or \( \mathbb{Z}_4 \).
- **B: \( U(8), \mathbb{Z}_4, \mathbb{Z}_2, \mathbb{Z}_1 \)** — This list contains all homomorphic images since \( U(8) \cong \mathbb{Z}_4 \).
- **C: \( U(8), \mathbb{Z}_2, \mathbb{Z}_1 \)** — This list is incomplete because it does not include \( \mathbb{Z}_4 \).
- **D: None of these** — This is incorrect because B contains all homomorphic images.

### Conclusion
The correct answer is **B**, as it contains all the homomorphic images of \( U(8) \), which are \( \mathbb{Z}_4 \), \( \mathbb{Z}_2 \), and \( \mathbb{Z}_1 \).

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Would you like more details or have any questions? 

Here are some related questions to expand on this topic:
1. What is the First Isomorphism Theorem, and how does it apply here?
2. How are homomorphic images related to quotient groups?
3. Why is \( U(8) \) isomorphic to \( \mathbb{Z}_4 \)?
4. How do we find the subgroups of \( \mathbb{Z}_4 \)?
5. What are some other examples of homomorphic images for different cyclic groups?

**Tip:** When analyzing homomorphic images, focus on understanding the structure of the group and its quotient groups, as these provide all possible homomorphic images.

Which of the following lists contains all homomorphic images of U(8) (up to isomorphism)? A: U(8), Z_1 B: U(8), Z_4, Z_2, Z_1 C: U(8), Z_2, Z_1 D: none of these contains all homomorphic images.

This problem explores the homomorphic images of U(8), the group of units modulo 8, using the First Isomorphism Theorem. Learn how to identify all homomorphic images up to isomorphism, suitable for undergraduate mathematics.

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