Problem 28 - Algebra: Pure and Applied

Question

In Exercises 23 through 30 construct an example of a nontrivial homomorphism between the two indicated groups, if this is possible, or explain why this is not possible. $$ \phi: \mathbb{Z} \rightarrow \mathbb{Z}_{7} $$

Step-by-Step Solution

Answer

A nontrivial homomorphism is \( \phi(n) = n \mod 7 \) with \( k = 1 \).

Step 1: Understand the Problem

We need to find a homomorphism \( \phi: \mathbb{Z} \rightarrow \mathbb{Z}_7 \). \( \mathbb{Z} \) is the group of integers under addition, and \( \mathbb{Z}_7 \) is the cyclic group of integers modulo 7 under addition.

Step 2: Properties of Homomorphism

A homomorphism from one group to another must preserve the group operation. Specifically, for our homomorphism \( \phi \), this means \( \phi(a+b) = \phi(a) + \phi(b) \) for all \( a, b \in \mathbb{Z} \).

Step 3: Identify a Suitable Map

To construct a nontrivial homomorphism, we need to choose \( \phi(1) = k \) where \( k \) is an integer between 0 and 6 in \( \mathbb{Z}_7 \). This ensures that \( \phi(n) = n \cdot k \mod 7 \) for any integer \( n \).

Step 4: Ensure the Homomorphism is Nontrivial

A nontrivial homomorphism cannot send every integer to 0 in \( \mathbb{Z}_7 \). Therefore, \( k eq 0 \). Let's choose \( k = 1 \), so the map becomes \( \phi(n) = n \mod 7 \). This is a valid nontrivial homomorphism.

Step 5: Verify the Homomorphism

Let's verify if \( \phi \) satisfies the homomorphism property. If \( \phi(n) = n \mod 7 \), then for any integers \( a \) and \( b \), \( \phi(a + b) = (a + b) \mod 7 = a \mod 7 + b \mod 7 \mod 7 = \phi(a) + \phi(b) \). This confirms it is a homomorphism.

Step 6: Analyze Other Possibilities

Other choices for \( k \) (such as 2, 3, 4, 5, 6) would also result in nontrivial homomorphisms. Each would define \( \phi(n) = n \cdot k \mod 7 \), and all satisfy the homomorphism property. But the critical factor is that \( k eq 0 \).

Key Concepts

Cyclic Groups

Let's delve into the fascinating world of cyclic groups, which play a central role in understanding various algebraic structures. A cyclic group is a group that can be generated by a single element. This means every member of the group can be written as a power (under group operation) of this generator. For example, the integers modulo 7, denoted as \(\mathbb{Z}_7\), forms a cyclic group under addition.

• Each element in \(\mathbb{Z}_7\) can be represented as \(0, 1, 2, 3, 4, 5, 6\).
• These elements cycle through themselves back to zero, creating a repeating pattern modulo 7. For instance, adding 5 and 4 in \(\mathbb{Z}_7\) yields 2: \((5 + 4) \mod 7 = 2\).
• This cyclic behavior, characteristic of groups like \(\mathbb{Z}_7\), enables a simplified structure with predictable patterns.

Cyclic groups are foundational in group theory because they provide a clearer perspective on how elements interact under the group operation. Furthermore, understanding them helps in constructing homomorphisms, which are mappings between groups that preserve structure.

Integers Modulo

When we talk about integers modulo \(n\), we're referring to a system where numbers "wrap around" after reaching a specific value, \(n\). Using our example, \(\mathbb{Z}_7\), every integer equates to one of the numbers 0 through 6.

• For any integer \(a\), \(a \mod 7\) gives the remainder when \(a\) is divided by 7.
• This operation has a "clock-like" cyclic nature, which makes it an ideal structure for modular arithmetic.
• A crucial aspect is that operations like addition within this framework are still intuitive, with watchful consideration of the modulus.

Modular arithmetic is widely applicable in fields such as cryptography and computer science. Its ability to reduce potentially infinite differences into manageable sets is particularly useful for understanding how homomorphisms like \(\phi: \mathbb{Z} \rightarrow \mathbb{Z}_7\) are constructed.

Nontrivial Homomorphism

A nontrivial homomorphism is essentially a function between two groups that maintains their structure yet does not map every element to the same single element, especially not the identity.

• In our case, the identity in \(\mathbb{Z}_7\) is 0, so a nontrivial homomorphism cannot map every integer \(n\) from \(\mathbb{Z}\) to 0 in \(\mathbb{Z}_7\).
• The mapping strategy involves selecting \(k\) from 1 to 6 in \(\mathbb{Z}_7\), since choosing 0 would make the homomorphism trivial.
• For example, setting \(k = 1\) results in \(\phi(n) = n \mod 7\), which is visibly nontrivial as it doesn’t collapse all elements to 0.
• This homomorphism satisfies the core property: \(\phi(a + b) = \phi(a) + \phi(b)\), validating its structural consistency between \(\mathbb{Z}\) and \(\mathbb{Z}_7\).

Understanding nontrivial homomorphisms is crucial because they showcase the connections and consistencies between different algebraic systems, revealing deeper insights into the groups' inherent properties.