Problem 5 - A Book of Abstract Algebra

Question

In each of the following, use the fundamental homomorphism theorem to prove that the two given groups are isomorphic. Then display their tables. \(\mathbb{Z}_{3}\) and \(\left(\mathbb{Z}_{3} \times \mathbb{Z}_{3}\right) / K\), where \(K=\\{(0,0),(1,1),(2,2)\\}\). [HINT: Consider the function \(f(a, b)=a-b\) from \(\mathbb{Z}_{3} \times \mathbb{Z}_{3}\) to \(\mathbb{Z}_{3}\).]

Step-by-Step Solution

Answer

The groups \( \mathbb{Z}_3 \) and \( (\mathbb{Z}_3 \times \mathbb{Z}_3) / K \) are isomorphic with group tables matching under isomorphism.

Step 1: Define the Homomorphism

Consider the function \( f: \mathbb{Z}_3 \times \mathbb{Z}_3 \rightarrow \mathbb{Z}_3 \) defined by the hint: \( f(a, b) = a - b \). This function takes a pair \((a, b)\) from \( \mathbb{Z}_3 \times \mathbb{Z}_3 \) and returns the result in \( \mathbb{Z}_3 \).

Step 2: Verify Homomorphism Properties

Check that \( f \) is a homomorphism. Compute \( f((a_1, b_1) + (a_2, b_2)) = f((a_1 + a_2, b_1 + b_2)) = (a_1 + a_2) - (b_1 + b_2) \). On the other hand, \( f(a_1, b_1) + f(a_2, b_2) = (a_1 - b_1) + (a_2 - b_2) = (a_1 + a_2) - (b_1 + b_2) \). Thus, \( f((a_1, b_1) + (a_2, b_2)) = f(a_1, b_1) + f(a_2, b_2) \).

Step 3: Identify Kernel and Image

Find the kernel of \( f \), which is \( \{(a, b) \mid f(a, b) = a - b = 0\} \). This implies \( a = b \), so \( \ker(f) = \{(0, 0), (1, 1), (2, 2)\} \), which is \( K \). The image of \( f \) is all of \( \mathbb{Z}_3 \) since \( f(a, 0) = a \) achieves all values in \( \mathbb{Z}_3 \).

Step 4: Apply the Fundamental Homomorphism Theorem

The fundamental homomorphism theorem states that \( (\mathbb{Z}_3 \times \mathbb{Z}_3) / \ker(f) \cong \operatorname{im}(f) \). Since \( \ker(f) = K \) and \( \operatorname{im}(f) = \mathbb{Z}_3 \), we have \( (\mathbb{Z}_3 \times \mathbb{Z}_3) / K \cong \mathbb{Z}_3 \). This demonstrates that the groups are isomorphic.

Step 5: Verify Group Tables

Construct the group table for \( \mathbb{Z}_3 \): it is \([0, 1, 2]\) under addition modulo 3.For \( (\mathbb{Z}_3 \times \mathbb{Z}_3) / K \), consider the cosets \( \{(0,0), (1,1), (2,2)\} + (a,b) \) which result effectively as \( (a, b) + K \). Since each element can be represented uniquely, distributing over these gives a similar result as \( \mathbb{Z}_3 \) addition, confirming the isomorphism.

Key Concepts

Group Isomorphism

Group Isomorphism is a fundamental concept in group theory where two groups are considered structurally the same, despite possible differences in their elements or operations. When two groups are isomorphic, there is a one-to-one correspondence between their elements that preserves group operations.

• To showcase isomorphism, we define a bijective (one-to-one and onto) function between two groups.
• This function must also maintain the operation structure of the groups. For example, if group operations are indicated by addition, then the function should satisfy the equation: \(f(a + b) = f(a) + f(b)\).

In the provided exercise, the groups \(\mathbb{Z}_{3}\) and \((\mathbb{Z}_{3} \times \mathbb{Z}_{3}) / K\) are shown to be isomorphic as the operation structure is retained between them. Ensuring isomorphism allows one to simplify complex problems by switching to a more familiar or simpler group while retaining all structural properties.

Kernel and Image

The kernel and image of a function are key concepts when working with homomorphisms, especially in the context of the Fundamental Homomorphism Theorem.

• The **kernel** of a homomorphism \(f: G \rightarrow H\) consists of all elements in \(G\) that are mapped to the identity element in \(H\). Mathematically, it is expressed as \(\ker(f) = \{g \in G \mid f(g) = e_H\}\).
• On the other hand, the **image** of a homomorphism is the set of all output values under the function. It is expressed as \(\operatorname{im}(f) = \{f(g) \mid g \in G\}\).

In the exercise, we define a homomorphism \(f(a, b) = a - b\), whose kernel is the subgroup \(K = \{(0, 0), (1, 1), (2, 2)\}\). The image of this map covers all possible output values in \(\mathbb{Z}_3\), playing a crucial role in proving that the two groups are isomorphic.

Modulo Arithmetic

Modulo Arithmetic, often referred to as clock arithmetic, is essential for understanding cyclic groups like \(\mathbb{Z}_3\). It is concerned with the remainder after division of one number by another.

• **Addition Modulo**: For any two integers \(a\) and \(b\), their addition modulo \(n\) is \((a + b) \mod n\). For \(\mathbb{Z}_3\), this means results are within the set \([0, 1, 2]\).
• This type of arithmetic ensures that calculations "wrap around" every \(n\) steps, similar to how hours on a clock reset after 12.

In the exercise, modulo arithmetic helps establish operations within \(\mathbb{Z}_3\) and within the cosets of \((\mathbb{Z}_3 \times \mathbb{Z}_3) / K\), showing that both have the same underlying cyclic structure.

Cosets

Cosets are an essential part of group theory. They allow us to partition groups into equal-size sets using a specific subgroup.

• A **left coset** of a subgroup \(H\) in a group \(G\) is formed by combining each element \(a\) of \(G\) with each element of \(H\), noted as \(aH = \{ah \mid h \in H\}\).
• Cosets determine whether a subgroup is normal, which implies that the left cosets are equivalent to right cosets.

In the context of the exercise, the groups formed by the cosets \((\mathbb{Z}_3 \times \mathbb{Z}_3) / K\) reflect the partitioning of \(\mathbb{Z}_3 \times \mathbb{Z}_3\) by the subgroup \(K = \ker(f)\). This visualization through cosets allows us to clearly map the structure and behaviour of the original group to a simpler, isomorphic representation, effective under modulo operation.