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 tackle each part of the problem one by one.

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

To show that \( f \) is a group homomorphism, we need to verify that for any \((a_1, b_1), (a_2, b_2) \in G\),

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

Given that \( G = H \times H \) and the operation in \( G \) is component-wise addition, we have:

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

Applying \( f \) to this result, we get:

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

On the other hand, applying \( f \) to each component separately and then adding the results gives:

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

Since addition in \( H = \mathbb{Z}/15\mathbb{Z} \) is associative and commutative, we have:

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

Therefore,

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

which confirms that \( f \) is a group homomorphism.

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

The kernel of a homomorphism \( f: G \to H \) is the set of all elements in \( G \) that map to the identity element in \( H \). The identity element in \( H = \mathbb{Z}/15\mathbb{Z} \) is 0. Therefore,

\[ \ker(f) = \{(a, b) \in G \mid f(a, b) = a + b = 0 \in \mathbb{Z}/15\mathbb{Z}\}. \]

In other words, 

\[ N = \{(a, b) \in H \times H \mid a + b \equiv 0 \pmod{15}\}. \]

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

To find the subgroups of the quotient group \( G/N \), we first need to understand the structure of \( G/N \). Since \( N = \ker(f) \), the First Isomorphism Theorem tells us that:

\[ G/N \cong \text{Im}(f). \]

The image of \( f \), denoted as \(\text{Im}(f)\), is:

\[ \text{Im}(f) = \{ f(a, b) \mid (a, b) \in G \} = \{ a + b \mid a, b \in \mathbb{Z}/15\mathbb{Z} \}. \]

Since \( a \) and \( b \) take all values in \( \mathbb{Z}/15\mathbb{Z} \), \( a + b \) will also take all values in \( \mathbb{Z}/15\mathbb{Z} \). Thus,

\[ \text{Im}(f) = \mathbb{Z}/15\mathbb{Z}. \]

So,

\[ G/N \cong \mathbb{Z}/15\mathbb{Z}. \]

The subgroups of \(\mathbb{Z}/15\mathbb{Z}\) are well-known and correspond to the divisors of 15. Therefore, the subgroups of \( \mathbb{Z}/15\mathbb{Z} \) (and hence of \( G/N \)) are isomorphic to:

1. \( \mathbb{Z}/15\mathbb{Z} \),
2. \( \mathbb{Z}/5\mathbb{Z} \),
3. \( \mathbb{Z}/3\mathbb{Z} \),
4. \( \{0\} \).

These correspond to the quotient groups of \( G/N \).

### Summary

1. The map \( f: G \to H \) defined by \( f(a, b) = a + b \) is a group homomorphism.
2. The kernel of \( f \) is \( N = \{(a, b) \in H \times H \mid a + b \equiv 0 \pmod{15}\} \).
3. The quotient group \( G/N \) is isomorphic to \( \mathbb{Z}/15\mathbb{Z} \), and its subgroups are isomorphic to \( \mathbb{Z}/15\mathbb{Z} \), \( \mathbb{Z}/5\mathbb{Z} \), \( \mathbb{Z}/3\mathbb{Z} \), and \( \{0\} \).

Do you want details or have any questions? Here are 8 related questions you might find interesting:

1. What is the First Isomorphism Theorem and how is it applied?
2. How do you determine the image of a group homomorphism?
3. What are the possible orders of elements in \( \mathbb{Z}/15\mathbb{Z} \)?
4. How are the subgroups of \( \mathbb{Z}/15\mathbb{Z} \) determined by its divisors?
5. What is the significance of the kernel in a group homomorphism?
6. How do you construct quotient groups and what are their properties?
7. How can the structure of \( G \times G \) affect the quotient group \( G/N \)?
8. Can you provide another example of a homomorphism and find its kernel and image?

**Tip:** Understanding the structure of quotient groups can greatly simplify the process of analyzing homomorphisms and their properties in group theory.

Explore group theory concepts involving homomorphisms, quotient groups, and the kernel of a homomorphism. Learn how to show a map between groups is a homomorphism, determine the kernel, and find subgroups of the quotient group. Suitable for advanced undergraduate students and beyond.

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