Let H = (Z/15Z, +), and let G = H × H.
1. Show that f : G → H defined by f(a, b) = a + b, for each a, b ∈ H, is a group
homomorphism.
2. Determine N = ker(f).
3. Determine (up to isomorphism) the subgroups of the quotient group G/N

Let's break down the problem step by step.

### 1. Show that \( f : G \to H \) defined by \( f(a, b) = a + b \), for each \( a, b \in H \), is a group homomorphism.

Given:
- \( H = (\mathbb{Z}/15\mathbb{Z}, +) \) is the additive group of integers modulo 15.
- \( G = H \times H \) is the direct product of \( H \) with itself, so elements of \( G \) are pairs \((a, b)\) where \( a, b \in \mathbb{Z}/15\mathbb{Z} \).
- \( f: G \to H \) is defined by \( f((a, b)) = a + b \).

To show that \( f \) is a group homomorphism, we need to prove that \( f \) preserves the group operation, i.e., for all \( (a_1, b_1), (a_2, b_2) \in G \),

\[
f((a_1, b_1) + (a_2, b_2)) = f((a_1, b_1)) + f((a_2, b_2)).
\]

#### Proof:
- The group operation in \( G \) is component-wise addition, so:
  \[
  (a_1, b_1) + (a_2, b_2) = (a_1 + a_2, b_1 + b_2).
  \]
- Applying \( f \) to this result:
  \[
  f((a_1 + a_2, b_1 + b_2)) = (a_1 + a_2) + (b_1 + b_2).
  \]
- On the other hand, the right-hand side of the homomorphism property is:
  \[
  f((a_1, b_1)) + f((a_2, b_2)) = (a_1 + b_1) + (a_2 + b_2).
  \]
- Since addition in \( \mathbb{Z}/15\mathbb{Z} \) is associative, we have:
  \[
  (a_1 + a_2) + (b_1 + b_2) = (a_1 + b_1) + (a_2 + b_2).
  \]
Thus, \( f((a_1, b_1) + (a_2, b_2)) = f((a_1, b_1)) + f((a_2, b_2)) \), proving that \( f \) is a group homomorphism.

### 2. Determine \( N = \ker(f) \).

The kernel of \( f \) is defined as:
\[
N = \ker(f) = \{(a, b) \in G \mid f(a, b) = 0_H\},
\]
where \( 0_H \) is the identity element in \( H \), which is \( 0 \) in \( \mathbb{Z}/15\mathbb{Z} \).

Since \( f(a, b) = a + b \), we have:
\[
N = \{(a, b) \in G \mid a + b = 0 \text{ in } \mathbb{Z}/15\mathbb{Z}\}.
\]
This means:
\[
N = \{(a, b) \mid a \equiv -b \pmod{15}\}.
\]
In other words, \( N \) consists of all pairs \((a, b)\) where \( b \equiv -a \pmod{15} \).

### 3. Determine (up to isomorphism) the subgroups of the quotient group \( G/N \).

The quotient group \( G/N \) consists of cosets of the form \( (a, b) + N \).

#### Structure of \( G/N \):
- \( G \) is isomorphic to \( \mathbb{Z}/15\mathbb{Z} \times \mathbb{Z}/15\mathbb{Z} \), which is a group of order \( 225 = 15 \times 15 \).
- \( N \) has the form \( \{(a, -a) \mid a \in \mathbb{Z}/15\mathbb{Z}\} \), and \( N \) is isomorphic to \( \mathbb{Z}/15\mathbb{Z} \).
- The quotient \( G/N \) therefore has order \( \frac{225}{15} = 15 \).

Since \( G/N \) is a quotient of a product of cyclic groups, and the quotient has order 15, \( G/N \) must be isomorphic to \( \mathbb{Z}/15\mathbb{Z} \).

Thus, \( G/N \cong \mathbb{Z}/15\mathbb{Z} \).

### Summary of Results:
1. \( f: G \to H \) is a group homomorphism.
2. \( \ker(f) = \{(a, b) \mid a + b = 0 \text{ in } \mathbb{Z}/15\mathbb{Z}\} \).
3. \( G/N \) is isomorphic to \( \mathbb{Z}/15\mathbb{Z} \).

---

Would you like further details on any of these steps or have any additional questions?

Here are some related questions:
1. How do you determine the order of an element in a group like \( \mathbb{Z}/15\mathbb{Z} \)?
2. What are the properties of a group homomorphism?
3. How do you determine if a given function between two groups is an isomorphism?
4. What are the possible subgroups of \( \mathbb{Z}/15\mathbb{Z} \)?
5. How do you find the cosets of a subgroup in a quotient group?
6. What is the significance of the kernel of a group homomorphism?
7. How is the order of a quotient group determined?
8. What is the First Isomorphism Theorem and how does it apply to quotient groups?

**Tip:** When dealing with quotient groups, understanding the kernel of the homomorphism can often simplify your analysis significantly.

Explore group theory concepts with a detailed analysis of homomorphism and quotient groups in the context of H = (Z/15Z) × (Z/15Z). Learn how to prove a group homomorphism, determine the kernel, and analyze the structure of the quotient group G/N. Suitable for advanced undergraduate studies in mathematics.

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