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

To solve the problem, we'll break it down step by step:

### Step 1: Understand the Group \( G \)
The group \( H = (\mathbb{Z}/15\mathbb{Z}, +) \) is the additive group of integers modulo 15. This group \( H \) is cyclic of order 15, generated by 1 (i.e., \( H = \langle 1 \rangle \)).

The group \( G \) is defined as \( G = H \times H \). Therefore, \( G \) is the direct product of two copies of \( H \), and it can be represented as \( G = (\mathbb{Z}/15\mathbb{Z}) \times (\mathbb{Z}/15\mathbb{Z}) \).

### Step 2: Structure of \( G \)
Since \( H \) is cyclic of order 15, \( G \) is a group of order \( 15 \times 15 = 225 \). \( G \) is also a product of two cyclic groups, so \( G \) itself is not cyclic but it is abelian.

### Step 3: Consider the Subgroups of \( G \)
To find the subgroups of the quotient group \( G/N \) (where \( N \) is some normal subgroup of \( G \)), we need to consider the structure of \( G/N \).

### Step 4: The Structure of \( G \times G \)
Let's consider a generic normal subgroup \( N \subseteq G \). Since \( G \) is abelian, all subgroups are normal, so we can consider \( N \) as any subgroup of \( G \).

### Step 5: Quotient Group \( G/N \)
The quotient group \( G/N \) will have the order equal to \( |G|/|N| \). The possible values for \( |N| \) are divisors of 225.

The divisors of 225 are:
\[ 1, 3, 5, 9, 15, 25, 45, 75, 225 \]

So, the possible orders of \( G/N \) are:
\[ 225, 75, 45, 25, 15, 9, 5, 3, 1 \]

### Step 6: Subgroups of the Quotient Group \( G/N \)
To determine the isomorphism classes of the subgroups of \( G/N \), we need to understand the structure of \( G/N \). If \( N \) has order \( k \), then \( G/N \) will be isomorphic to some group of order \( 225/k \). The possible quotient groups are:

- If \( |N| = 1 \), \( G/N \cong G \), so subgroups correspond to the subgroups of \( G \), which is isomorphic to \( (\mathbb{Z}/15\mathbb{Z}) \times (\mathbb{Z}/15\mathbb{Z}) \).

- If \( |N| = 3 \), \( G/N \cong (\mathbb{Z}/5\mathbb{Z}) \times (\mathbb{Z}/15\mathbb{Z}) \).

- If \( |N| = 5 \), \( G/N \cong (\mathbb{Z}/3\mathbb{Z}) \times (\mathbb{Z}/15\mathbb{Z}) \).

- If \( |N| = 9 \), \( G/N \cong (\mathbb{Z}/5\mathbb{Z}) \times (\mathbb{Z}/3\mathbb{Z}) \).

- If \( |N| = 15 \), \( G/N \cong (\mathbb{Z}/3\mathbb{Z}) \times (\mathbb{Z}/5\mathbb{Z}) \).

- If \( |N| = 25 \), \( G/N \cong (\mathbb{Z}/9\mathbb{Z}) \).

- If \( |N| = 45 \), \( G/N \cong (\mathbb{Z}/5\mathbb{Z}) \).

- If \( |N| = 75 \), \( G/N \cong (\mathbb{Z}/3\mathbb{Z}) \).

- If \( |N| = 225 \), \( G/N \cong \{0\} \).

### Step 7: Isomorphism Classes of Subgroups
The subgroups of the quotient groups \( G/N \) (up to isomorphism) are as follows:

- **Order 1**: Trivial group.
- **Order 3**: \( \mathbb{Z}/3\mathbb{Z} \).
- **Order 5**: \( \mathbb{Z}/5\mathbb{Z} \).
- **Order 9**: \( \mathbb{Z}/9\mathbb{Z} \).
- **Order 15**: \( \mathbb{Z}/15\mathbb{Z} \).
- **Order 25**: \( \mathbb{Z}/25\mathbb{Z} \).
- **Order 45**: \( \mathbb{Z}/45\mathbb{Z} \).
- **Order 75**: \( \mathbb{Z}/75\mathbb{Z} \).
- **Order 225**: \( G \) itself.

Thus, these are the possible isomorphism classes of the subgroups of the quotient groups \( G/N \) where \( N \) is a normal subgroup of \( G \).

### Conclusion
The subgroups of the quotient group \( G/N \) (up to isomorphism) can be isomorphic to:
\[ \mathbb{Z}/3\mathbb{Z}, \mathbb{Z}/5\mathbb{Z}, \mathbb{Z}/9\mathbb{Z}, \mathbb{Z}/15\mathbb{Z}, \mathbb{Z}/25\mathbb{Z}, \mathbb{Z}/45\mathbb{Z}, \mathbb{Z}/75\mathbb{Z}, \mathbb{Z}/225\mathbb{Z} \text{ (the trivial group)}. \]

Would you like more details on any of these points? Here are some related questions:

1. What is the significance of the group \( G = H \times H \) being abelian?
2. How does the order of a subgroup \( N \) relate to the structure of \( G/N \)?
3. Why are all subgroups of \( G \) normal?
4. How do the isomorphism classes change if \( H \) were not cyclic?
5. Can you describe the process of determining the isomorphism type of a quotient group?
6. How can you generalize this approach to other cyclic groups?
7. What would happen if \( G \) were non-abelian?
8. How do you find all subgroups of \( (\mathbb{Z}/15\mathbb{Z}) \times (\mathbb{Z}/15\mathbb{Z}) \)?

**Tip:** When working with quotient groups, always consider the orders of the groups and subgroups first to understand the possible structures involved.

Explore the subgroups of the quotient group G/N where G = (Z/15Z) x (Z/15Z). Understand the structure of G as a direct product of cyclic groups, and determine the isomorphism classes of its subgroups. Suitable for advanced undergraduate students and researchers in group theory.

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