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 \(\phi\). $$ \text { \phi: } \mathbb{Z}_{6} \rightarrow \mathbb{Z}_{2} \text { , where } \phi(x)=\text { the remainder of } x \text { mod } 2 $$

Short answer

The map \( \phi \) is a homomorphism with \( \ker \phi = \{0, 2, 4\} \).

Step-by-step solution

Understanding the Problem

We need to determine if the given map \( \phi : \mathbb{Z}_{6} \to \mathbb{Z}_{2} \), where \( \phi(x) \) is the remainder of \( x \mod 2 \), is a homomorphism. Then, if it is, we'll find \( \ker \phi \).

Define the Map

The map \( \phi(x) = x \mod 2 \) takes an integer \( x \in \mathbb{Z}_{6} \) and maps it to its remainder when divided by 2, giving a result in \( \mathbb{Z}_{2} \), which contains elements \{0, 1\}.

Check Homomorphism Property

A map is a homomorphism if \( \phi(a + b) = \phi(a) + \phi(b) \) for all \( a, b \in \mathbb{Z}_6 \). Check this by selecting arbitrary elements and verifying this property.

Verify with Elements

Consider elements in \( \mathbb{Z}_6 \), such as \( a = 3 \) and \( b = 4 \):\\( \phi(3) = 3 \mod 2 = 1 \) and \( \phi(4) = 4 \mod 2 = 0 \).\\( \phi(3 + 4) = \phi(7 \mod 6) = 1 \) and \( \phi(3) + \phi(4) = 1 + 0 = 1 \).\This holds true: \( \phi(a+b) = \phi(a) + \phi(b) \). Repeat this for other combinations to confirm.

Conclusion about Homomorphism

The map \( \phi \) satisfies the homomorphism condition for all elements, so \( \phi \) is indeed a homomorphism from \( \mathbb{Z}_{6} \to \mathbb{Z}_{2} \).

Determine the Kernel

The kernel \( \ker \phi \) consists of all elements \( x \in \mathbb{Z}_6 \) such that \( \phi(x) = 0 \). Solve \( x \mod 2 = 0 \), which gives the set \( \{0, 2, 4\} \).

Finalize Kernel Determination

Thus, \( \ker \phi = \{0, 2, 4\} \) comprises those elements of \( \mathbb{Z}_6 \) that map to 0 in \( \mathbb{Z}_2 \).

Key concepts

Kernel of a Homomorphism

When exploring the concept of a homomorphism, the kernel plays an intriguing role. It's essentially the soul of the map, capturing all the elements that are sent to the neutral element of the codomain. In the context of group homomorphisms, the kernel consists of the elements that the homomorphism maps to the identity element in the target group.

This brings us to the kernel of the homomorphism \( \phi: \mathbb{Z}_6 \rightarrow \mathbb{Z}_2 \). The function \( \phi(x) = x \mod 2 \) sends elements to 0 or 1 in \( \mathbb{Z}_2 \). Thus, the kernel \( \ker \phi \) are those elements in \( \mathbb{Z}_6 \) which map to 0 under \( \phi \).

To find it, solve \( x \mod 2 = 0 \), leading to \( \ker \phi = \{0, 2, 4\} \). These are the elements that, when divided by 2, leave no remainder. Understanding the kernel helps us know how much of the original structure collapses into a simpler form under the map.

Modular Arithmetic

Modular arithmetic is like wrapping numbers around a circle, where numbers reset after reaching a certain value, known as the modulus. It simplifies calculations and is sometimes called "clock arithmetic" because of how numbers circle back around similarly to hours on a clock.

In this exercise, we're working with \( x \mod 2 \). The operation divides an integer \( x \) by 2 and gives us the remainder. For example, \( 3 \mod 2 = 1 \) because 3 divided by 2 leaves a remainder of 1. Similarly, \( 4 \mod 2 = 0 \) since 4 is perfectly divisible by 2 without any remainder.

Modular arithmetic is a cornerstone in many areas of math and computer science, particularly in number theory and cryptography, because it allows the manipulation of numbers within a confined set.

Group Theory

Group theory is a fascinating branch of mathematics exploring sets with an operation that combines any two elements to form a third element, while also satisfying four essential properties: closure, associativity, the presence of an identity element, and the existence of inverse elements.

In our context, we're dealing with the groups \( \mathbb{Z}_6 \) and \( \mathbb{Z}_2 \), consisting of integers under the operation of addition modulo 6 and 2, respectively. These groups demonstrate how mathematical objects can be combined and transformed while obeying specific rules.

Studying transformations between groups, known as homomorphisms, offers profound insights into the group's structure and functions. The map \( \phi: \mathbb{Z}_6 \to \mathbb{Z}_2 \) discussed here is a homomorphism, showing a structured way integers can be converted and simplified while preserving inherent arithmetic properties.

Integers Modulo n

Integers modulo \( n \), denoted \( \mathbb{Z}_n \), form a set of equivalence classes under the relation of congruence modulo \( n \). This means two numbers are considered the same if they leave the same remainder when divided by \( n \).

For instance, in \( \mathbb{Z}_6 \), numbers like \( 0, 6, \) and \( 12 \) are all equivalent because they all leave a remainder of 0 when divided by 6. This modular world helps us see patterns and relationships that are not as apparent when dealing with raw integers.

The operation of taking numbers "mod \( n \)" allows arithmetic to be performed within a fixed range, simplifying many calculations. This is integral in solving problems in cryptography, coding theory, and even in everyday applications like hashing in computer algorithms.