Question

In Exercises 1 through \(10,\) determine whether or not the indicated map \(\phi\) is a homomorphism, and in the cases where \(\phi\) is a homomorphism, determine Kem \(\phi\). $$ \phi: \mathbb{Z}_{7} \rightarrow \mathbb{Z}_{2}, \text { where } \phi(x)=\text { the remainder of } x \text { mod } 2 $$

Short answer

Yes, \( \phi \) is a homomorphism. The kernel is \( \{0, 2, 4, 6\} \).

Step-by-step solution

Understanding Homomorphism

A map \( \phi: G \rightarrow H \) is a homomorphism between two groups \( G \) and \( H \) if for any elements \( a, b \in G \), \( \phi(ab) = \phi(a)\phi(b) \). For this exercise, we need to check if the map \( \phi: \mathbb{Z}_{7} \rightarrow \mathbb{Z}_{2} \), defined as \( \phi(x) = x \mod 2 \), satisfies this property.

Checking Homomorphism Property

Consider any \( a, b \in \mathbb{Z}_{7} \). Calculate \( \phi(a+b) = (a+b) \mod 2 \) and compare it to \( \phi(a) + \phi(b) = (a \mod 2) + (b \mod 2) \mod 2 \). For the homomorphism condition to hold, both expressions need to be equal.

Verifying with Examples

Let's check a few cases: 1. If \( a = 2 \) and \( b = 3 \), then \( (a+b) \mod 2 = 5 \mod 2 = 1 \) and \( (a \mod 2) + (b \mod 2) = (0 + 1) \mod 2 = 1 \).2. If \( a = 4 \) and \( b = 3 \), then \( (a+b) \mod 2 = 7 \mod 2 = 1 \) and \( (a \mod 2) + (b \mod 2) = (0 + 1) \mod 2 = 1 \).In both cases \( \phi(a+b) = \phi(a) + \phi(b) \), indicating \( \phi \) is a homomorphism.

Determining the Kernel

The kernel of a homomorphism \( \phi: G \rightarrow H \) is the set \( \text{Ker } \phi = \{ g \in G \mid \phi(g) = e_H \} \), where \( e_H \) is the identity element of \( H \). Here, the identity in \( \mathbb{Z}_2 \) is 0. So, \( \text{Ker } \phi = \{ x \in \mathbb{Z}_7 \mid x \mod 2 = 0 \} \).

Finding Elements in the Kernel

In \( \mathbb{Z}_7 \), the elements are \( 0, 1, 2, 3, 4, 5, 6 \). The elements for which \( x \mod 2 = 0 \) are \( 0, 2, 4, 6 \), all of which map to 0 under \( \phi \). Thus, \( \text{Ker } \phi = \{0, 2, 4, 6\} \).

Key concepts

Group Theory

Group theory is the mathematical study of groups, which are algebraic structures used to model symmetry and other mathematical concepts. A group is defined as a set, equipped with a binary operation that combines two elements to form another element uniquely. This operation must satisfy four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverses for each element. These properties ensure that any algebraic manipulations performed in the group adhere to consistent rules. A homomorphism is a function between two groups that respects the group operation, meaning that the image of any combination of elements is the combination of their images. Homomorphisms allow us to explore the relationships between different group structures, capturing their internal symmetries. Understanding these concepts helps solve many mathematical and real-world problems, as they provide a way to reason about abstract structures and their properties.

Kernel of a Homomorphism

The kernel of a homomorphism plays a crucial role in understanding the structure of a group homomorphism. If you have a homomorphism \( \phi: G \rightarrow H \), the kernel is defined as the set of all elements in \( G \) that map to the identity element in \( H \). Mathematically, this is expressed as:

• \( \text{Ker } \phi = \{ g \in G : \phi(g) = e_H \} \)

Identifying the kernel helps us to determine the properties and the nature of a homomorphism. If the kernel only contains the identity element of \( G \), the homomorphism is called injective, meaning it maps distinct elements of \( G \) to distinct elements of \( H \).In the context of the exercise, since we are mapping from \( \mathbb{Z}_7 \) to \( \mathbb{Z}_2 \) and the identity of \( \mathbb{Z}_2 \) is 0, the kernel becomes the set of elements that are congruent to 0 modulo 2. Understanding kernels is a foundational step towards the exploration of quotient groups and more advanced group theory.

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, which considers the remainder when one integer is divided by another. It is sometimes referred to as 'clock arithmetic' because it cycles back to the beginning after reaching a certain value, much like a clock resets after 12 hours.In modular arithmetic, two numbers are considered equivalent if they have the same remainder when divided by the modulus. The notation \( a \equiv b \pmod{n} \) is used to indicate that the difference \( a - b \) is divisible by \( n \).In the exercise, we deal with \( x \mod 2 \), where any integer \( x \) is divided by 2, and only the remainder is considered. For example:

• If \( x = 5 \), then \( x \mod 2 = 1 \).
• If \( x = 4 \), then \( x \mod 2 = 0 \).

This operation is simple yet powerful, serving as the basis for various fields ranging from cryptography to computer science, and understanding it is essential for grasping how group homomorphisms in modular settings work.