Problem 4 - A Book of Abstract Algebra

Question

If \(n\) is a multiple of \(m\), then \(\mathbb{Z}_{m}\) is a homomorphic image of \(\mathbb{Z}_{n}\).

Step-by-Step Solution

Answer

\(\mathbb{Z}_m\) is a homomorphic image of \(\mathbb{Z}_n\) via a surjective homomorphism.

Step 1: Define the Groups

We need to understand the groups involved first, which are \(\mathbb{Z}_n\) and \(\mathbb{Z}_m\). \(\mathbb{Z}_n\) is a group of integers under addition modulo \(n\), and \(\mathbb{Z}_m\) is a group of integers under addition modulo \(m\).

Step 2: Establish the Relationship Between\(n\) and \(m\)

Since \(n\) is a multiple of \(m\), we can express \(n = km\) for some integer \(k\). This shows a relationship between \(\mathbb{Z}_n\) and \(\mathbb{Z}_m\).

Step 3: Define the Natural Homomorphism

Define a function \(\phi : \mathbb{Z}_n \to \mathbb{Z}_m\) by \(\phi([a]_n) = [a]_m\), where \([a]_n\) is the equivalence class of \(a\) in \(\mathbb{Z}_n\). This function maps elements from one group to the other.

Step 4: Verify Homomorphism Property

We must prove that \(\phi\) is a homomorphism, meaning \(\phi([a]_n + [b]_n) = \phi([a]_n) + \phi([b]_n)\). Calculate the left side: \(\phi([a+b]_n) = [a+b]_m\), and the right side: \(\phi([a]_n) + \phi([b]_n) = [a]_m + [b]_m = [a+b]_m\). The left side equals the right side, confirming \(\phi\) is a homomorphism.

Step 5: Show Surjectivity

Show that \(\phi\) is surjective, meaning every element in \(\mathbb{Z}_m\) has a preimage in \(\mathbb{Z}_n\). Take any \([b]_m \in \mathbb{Z}_m\). Since \(n\) is a multiple of \(m\), such a \(b\) will always have a corresponding preimage \([b]_n\) in \(\mathbb{Z}_n\).

Step 6: Conclude the Homomorphic Image

Since \(\phi\) is both a homomorphism and surjective, \(\mathbb{Z}_m\) is a homomorphic image of \(\mathbb{Z}_n\). This means \(\mathbb{Z}_m\) can be derived from \(\mathbb{Z}_n\) via a homomorphism.

Key Concepts

Group Theory

Group Theory is a branch of algebra that studies collections of objects, that can be combined in certain ways. These collections are known as "groups." The concept of a group is crucial to many areas in mathematics because it captures the idea of symmetrical operations.
Understanding groups involves looking at two things: a set of elements and a binary operation. This operation must satisfy four main properties:

• **Closure**: Performing the operation on two elements from the set produces another element within the same set.
• **Associativity**: Changing the grouping of elements does not change the result. For example, \((a + b) + c = a + (b + c)\).
• **Identity**: There exists an element within the set that acts as an identity element, so applying the operation with this element leaves any element unchanged.
• **Inverses**: For every element, there is another element in the set such that the operation yields the identity element.

Group Theory forms the backbone of many mathematical fields and is pivotal in understanding homomorphic images in algebra.

Modulo Arithmetic

Modulo Arithmetic, sometimes called clock arithmetic, deals with numbers when divided by a fixed divisor. The result is the remainder when one integer is divided by another. In other words, we study what remains after division.
In the context of groups like \(\mathbb{Z}_n\), elements are considered equivalent if they share the same remainder when divided by \(n\). Mathematically, this is expressed as \([a]_n\), where \(a\) is an integer and \([a]_n\) denotes its equivalence class in modulo \(n\).
Modulo arithmetic is foundational to understanding groups like \(\mathbb{Z}_n\) and \(\mathbb{Z}_m\), where arithmetic operations are performed under a modulus. It simplifies complex operations and is essential in forming the link between number theory and abstract algebra.

Surjective Homomorphism

A surjective homomorphism is a special type of function that plays a crucial role in algebra. Homomorphisms are functions that respect the structure between two algebraic objects, ensuring operations are preserved. For homomorphisms, if two elements in the first structure are combined, their images in the second structure should combine in the same way.
Surjectivity implies that the homomorphism covers every element in the target set. That is, for every element in the target group \(\mathbb{Z}_m\), there exists at least one corresponding element in the source group \(\mathbb{Z}_n\).
Understanding surjective homomorphisms is essential when dealing with concepts like homomorphic images, because it confirms that all elements of one structure can be reached from another, ensuring a complete mapping between the components.

Equivalence Classes

Equivalence Classes group elements that share a specific equivalence relation, simplifying complex collections of elements into simplified sets. In terms of modulo arithmetic, elements belong to the same equivalence class if they yield the same remainder when divided by a modulus.
Formally, if two elements \(a\) and \(b\) are in the same equivalence class modulo \(n\), we write \(a \equiv b \pmod{n}\). This notation indicates that \(a\) and \(b\) leave the same remainder when divided by \(n\).
Equivalence classes are instrumental in algebra, as they provide a straightforward way to categorize and work with infinite sets of numbers, reducing them to manageable groups containing elements that behave similarly under certain operations. This concept allows a deeper understanding of the relationships between different groups.