Let H = (Z/15Z, +), and let G = H × H.
Determine (up to isomorphism) the subgroups of the quotient group G/N

To determine the subgroups of the quotient group \( G/N \), where \( G = H \times H \) and \( H = (\mathbb{Z}/15\mathbb{Z}, +) \), we need to break down the problem systematically. Here's the detailed process:

### Step 1: Understand \( H = \mathbb{Z}/15\mathbb{Z} \)
The group \( H = \mathbb{Z}/15\mathbb{Z} \) is the cyclic group of order 15. The structure of \( H \) can be understood by decomposing it into its prime factor components:
\[
\mathbb{Z}/15\mathbb{Z} \cong \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z}
\]
So, \( H \cong \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z} \).

### Step 2: Structure of \( G = H \times H \)
Given \( G = H \times H \), we have:
\[
G \cong (\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z}) \times (\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z})
\]
This simplifies to:
\[
G \cong (\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}) \times (\mathbb{Z}/5\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z})
\]
Therefore, \( G \) is isomorphic to the direct product of two Klein four-groups (\( \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \)) and two cyclic groups of order 25 (\( \mathbb{Z}/5\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z} \)).

### Step 3: The Quotient Group \( G/N \)
The next step is to understand \( N \), the normal subgroup of \( G \). Since \( N \) is not specified, we assume \( N \) could be any normal subgroup of \( G \).

1. **Identify possible normal subgroups \( N \):** 
   - If \( N \) corresponds to a direct product of subgroups of \( \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \) and \( \mathbb{Z}/5\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z} \), then \( N \) could correspond to any combination of the factors.
   - Consider cases where \( N \) is trivial or non-trivial but still normal.

2. **Determine the structure of \( G/N \):** 
   - If \( N \) is trivial (i.e., \( N = \{(0, 0)\} \)), then \( G/N \cong G \).
   - If \( N \) corresponds to a subgroup that factors out some cyclic part, you will have a quotient group that is isomorphic to a smaller direct product of cyclic groups.

### Step 4: Determine the Subgroups of \( G/N \) up to Isomorphism
The subgroups of a quotient group \( G/N \) will correspond to the subgroups of \( G \) that contain \( N \). If \( G \cong (\mathbb{Z}/3\mathbb{Z})^2 \times (\mathbb{Z}/5\mathbb{Z})^2 \), the subgroups are generated by combinations of the cyclic groups.

- **For \( \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \):**
  - The possible subgroups are the trivial subgroup, the entire group, and the two cyclic subgroups generated by one of the elements (each of order 3).
  - So, there are 4 subgroups in total: \( \{(0,0)\}, \mathbb{Z}/3\mathbb{Z} \times \{0\}, \{0\} \times \mathbb{Z}/3\mathbb{Z}, \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \).

- **For \( \mathbb{Z}/5\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z} \):**
  - Similarly, there are 4 subgroups: \( \{(0,0)\}, \mathbb{Z}/5\mathbb{Z} \times \{0\}, \{0\} \times \mathbb{Z}/5\mathbb{Z}, \mathbb{Z}/5\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z} \).

### Summary
The subgroups of \( G/N \) will depend on the structure of \( N \). However, up to isomorphism, the subgroups will be combinations of the form:
- Subgroups of \( \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \),
- Subgroups of \( \mathbb{Z}/5\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z} \),
- Direct products of these subgroups.

These subgroups correspond to various direct sums of cyclic groups like \( \mathbb{Z}/3\mathbb{Z} \), \( \mathbb{Z}/5\mathbb{Z} \), and products of these forms.

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Would you like more details on specific subgroups, or do you have other questions?

Here are some related questions you might find interesting:
1. How do you compute the order of elements in a direct product of cyclic groups?
2. What is the fundamental theorem of finite abelian groups?
3. Can you describe the isomorphism types of groups of a given order?
4. How do you find all the normal subgroups of a given group?
5. What are some examples of non-trivial normal subgroups in direct product groups?
6. What is the relationship between quotient groups and subgroups?
7. Can you explain the concept of group homomorphisms and their kernels?
8. How does the Lattice Theorem apply to subgroups of quotient groups?

**Tip:** When dealing with quotient groups, always check how the normal subgroup \( N \) interacts with the structure of the group \( G \), as this will heavily influence the resulting quotient group.

Explore the subgroups of the quotient group G/N where G = (Z/15Z) x (Z/15Z) and H = Z/15Z. Understand the structure of G, identify normal subgroups, and determine the isomorphism types of subgroups. Detailed solution based on group theory concepts.

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