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. We are given a group \( G = H \times H \), where \( H = \mathbb{Z}/15\mathbb{Z} \) under addition. We need to analyze the function \( f: G \rightarrow H \) defined by \( f(a, b) = a + b \), where \( a, b \in H \).

### Part 1: Proving that \( f \) is a Group Homomorphism

A map \( f: G \rightarrow H \) is a group homomorphism if for all \( (a_1, b_1), (a_2, b_2) \in G \), the following holds:
\[
f((a_1, b_1) + (a_2, b_2)) = f(a_1, b_1) + f(a_2, b_2).
\]

First, note that in \( G \), the addition operation is defined as:
\[
(a_1, b_1) + (a_2, b_2) = (a_1 + a_2, b_1 + b_2),
\]
where \( + \) is addition modulo 15.

Now, let’s compute each side of the homomorphism condition:

1. **Left-hand side:**
   \[
   f((a_1, b_1) + (a_2, b_2)) = f(a_1 + a_2, b_1 + b_2).
   \]
   By definition of \( f \):
   \[
   f(a_1 + a_2, b_1 + b_2) = (a_1 + a_2) + (b_1 + b_2).
   \]
   In \( H = \mathbb{Z}/15\mathbb{Z} \), this sum is taken modulo 15.

2. **Right-hand side:**
   \[
   f(a_1, b_1) + f(a_2, b_2) = (a_1 + b_1) + (a_2 + b_2).
   \]
   Again, the sum is taken modulo 15.

Notice that both sides are equal because addition in \( \mathbb{Z}/15\mathbb{Z} \) is associative and commutative. Therefore, \( f \) is indeed a homomorphism.

### Part 2: Determining \( N = \ker(f) \)

The kernel of \( f \) is defined as:
\[
N = \ker(f) = \{ (a, b) \in G \mid f(a, b) = 0 \}.
\]
This means we need to find all pairs \( (a, b) \in \mathbb{Z}/15\mathbb{Z} \times \mathbb{Z}/15\mathbb{Z} \) such that \( a + b \equiv 0 \pmod{15} \).

In other words, \( a \equiv -b \pmod{15} \). The elements of \( N \) are:
\[
N = \{ (a, -a) \mid a \in \mathbb{Z}/15\mathbb{Z} \}.
\]

There are 15 such elements since \( a \) can take any value in \( \mathbb{Z}/15\mathbb{Z} \). So,
\[
N = \{ (0, 0), (1, 14), (2, 13), \dots, (14, 1) \}.
\]

### Part 3: Subgroups of the Quotient Group \( G/N \)

The quotient group \( G/N \) consists of cosets of \( N \) in \( G \). Each coset is of the form:
\[
(a, b) + N \text{ for } (a, b) \in G.
\]

To understand the structure of \( G/N \), let’s determine the order of the quotient group. The order of \( G \) is \( 15 \times 15 = 225 \). The order of \( N \) is 15, so the order of \( G/N \) is:
\[
\frac{225}{15} = 15.
\]

The quotient group \( G/N \) is isomorphic to \( \mathbb{Z}/15\mathbb{Z} \). This is because \( G/N \) has 15 elements, and the operation is addition modulo 15.

### Conclusion

- \( f \) is a group homomorphism.
- \( \ker(f) \) is \( \{ (a, -a) \mid a \in \mathbb{Z}/15\mathbb{Z} \} \).
- The quotient group \( G/N \) is isomorphic to \( \mathbb{Z}/15\mathbb{Z} \).

### Follow-Up Questions

1. What are the possible orders of subgroups of \( G/N \)?
2. Can you describe the cyclic subgroup structure within \( G/N \)?
3. How would the structure change if \( H = \mathbb{Z}/n\mathbb{Z} \) for some \( n \neq 15 \)?
4. What is the center of the group \( G = H \times H \)?
5. How would you generalize this analysis for \( H \times H \times H \)?

### Tip

When determining the quotient group \( G/N \), always start by calculating the orders of \( G \) and \( N \) to understand the size and structure of the resulting quotient group.

Explore the concept of group homomorphisms and quotient groups in group theory. Understand how the function f: G → H defined by f(a, b) = a + b for groups G = H × H, where H = Z/15Z, forms a homomorphism. Learn to determine the kernel of f and analyze subgroups of the quotient group G/N.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation