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.

To prove that \( f: G \to H \), defined by \( f((a, b)) = a + b \) for each \( a, b \in H \), is a group homomorphism, we need to verify the following:

1. **Function Definition:** \( f \) is well-defined, meaning it maps elements from \( G \) to \( H \).
2. **Preservation of the Group Operation:** \( f \) respects the group operation, i.e., \( f((a_1, b_1) + (a_2, b_2)) = f((a_1, b_1)) + f((a_2, b_2)) \) for all \( (a_1, b_1), (a_2, b_2) \in G \).

### Step 1: Function Definition
- \( G = H \times H \), where \( H = (\mathbb{Z}/15\mathbb{Z}, +) \).
- Elements of \( G \) are pairs \( (a, b) \), where \( a, b \in \mathbb{Z}/15\mathbb{Z} \).

The function \( f \) is defined as \( f: G \to H \) by \( f((a, b)) = a + b \).

Since \( a + b \in \mathbb{Z}/15\mathbb{Z} \), \( f((a, b)) \) is indeed an element of \( H \). Therefore, \( f \) is a well-defined function from \( G \) to \( H \).

### Step 2: Preservation of the Group Operation
We need to verify that \( f \) preserves the group operation. In other words, for any \( (a_1, b_1), (a_2, b_2) \in G \), we must show that:

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

#### Group Operation in \( G \):
The group operation in \( G = H \times H \) is defined as component-wise addition:

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

#### Applying \( f \) to \( (a_1 + a_2, b_1 + b_2) \):
\[
f((a_1 + a_2, b_1 + b_2)) = (a_1 + a_2) + (b_1 + b_2)
\]

#### Applying \( f \) to \( (a_1, b_1) \) and \( (a_2, b_2) \) separately:
\[
f((a_1, b_1)) = a_1 + b_1
\]
\[
f((a_2, b_2)) = a_2 + b_2
\]

#### Adding the results of \( f((a_1, b_1)) \) and \( f((a_2, b_2)) \):
\[
f((a_1, b_1)) + f((a_2, b_2)) = (a_1 + b_1) + (a_2 + b_2)
\]

Since addition in \( \mathbb{Z}/15\mathbb{Z} \) is associative, we can rewrite this as:
\[
(a_1 + b_1) + (a_2 + b_2) = (a_1 + a_2) + (b_1 + b_2)
\]

This shows that:
\[
f((a_1, b_1) + (a_2, b_2)) = f((a_1, b_1)) + f((a_2, b_2))
\]

### Conclusion
Since \( f \) preserves the group operation, it is a group homomorphism from \( G \) to \( H \).

Would you like any additional details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How do you prove that a function is a group isomorphism?
2. What are the properties of a group homomorphism?
3. How do you find the kernel of a group homomorphism?
4. What is the image of the homomorphism \( f \) in this context?
5. How does the structure of \( \mathbb{Z}/15\mathbb{Z} \) influence the behavior of the homomorphism \( f \)?

**Tip:** When proving that a function is a group homomorphism, it's crucial to carefully check the preservation of the group operation for all elements in the domain.

Learn how to prove that the function f: G → H, defined by f((a, b)) = a + b for each a, b ∈ H, is a group homomorphism. This problem involves elements from group theory, specifically focusing on group operations and homomorphisms. Suitable for advanced undergraduate students.

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