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. $$ \phi: \mathbb{Z}_{5} \rightarrow \mathbb{Z}_{12} $$

Short answer

No nontrivial homomorphism exists from \( \mathbb{Z}_{5} \) to \( \mathbb{Z}_{12} \).

Step-by-step solution

Understand Group Properties

The problem involves two groups: \( \mathbb{Z}_{5} \) and \( \mathbb{Z}_{12} \). \( \mathbb{Z}_{5} \) is the group of integers under addition modulo 5, and \( \mathbb{Z}_{12} \) is the group of integers under addition modulo 12.

Define a Homomorphism

A homomorphism \( \phi: \mathbb{Z}_{5} \rightarrow \mathbb{Z}_{12} \) is a function that respects the group operation, meaning \( \phi(a + b) = \phi(a) + \phi(b) \) for any integers \( a, b \). The identity element of \( \mathbb{Z}_{5} \) maps to the identity element of \( \mathbb{Z}_{12} \).

Check for a Trivial Homomorphism

The trivial homomorphism maps all elements of \( \mathbb{Z}_{5} \) to zero in \( \mathbb{Z}_{12} \). However, we seek a nontrivial homomorphism that maps some elements of \( \mathbb{Z}_{5} \) to non-zero elements in \( \mathbb{Z}_{12} \).

Investigate Order Compatibility

The order of \( \mathbb{Z}_{5} \) is 5. Given \( \phi(1) = k \), the kernel condition requires \( 5k \equiv 0 \mod 12 \). Solving this, \( 5k = 12m \) leads to \( k = \frac{12m}{5} \), not an integer, implying no nontrivial solution exists.

Conclude on Homomorphism Possibility

Since 5 and 12 are coprime, \( \phi(1) = k \) cannot simultaneously satisfy the condition \( 5k \equiv 0 \mod 12 \) without \( k = 0 \). Hence, the only homomorphism is the trivial one and no nontrivial homomorphism exists.

Key concepts

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value, called the modulus. It's like the numbers on a clock that reset after reaching 12. When you perform operations like addition or multiplication, the results are taken modulo a number, which is a way of saying that you divide by that number and take the remainder.

For example, in modular arithmetic with a modulus of 5, the set of numbers is {0, 1, 2, 3, 4}. When you add 3 and 2, instead of getting 5, you will get 0, because 5 modulo 5 equals 0. This is crucial in group theory, where we are often interested in the properties of operations under some modulus.

• Modulo: The operation of finding the remainder after division of one number by another.
• Set: A collection of numbers that follows the rules of modular arithmetic.
• Wrap around: Resetting to the start of the range once reaching the modulus.

Group Theory

Group theory is the study of algebraic structures known as groups. A group is a set equipped with an operation that combines any two of its elements to form a third element and satisfies four fundamental properties: closure, associativity, identity and invertibility.

In group theory, the elements of the group, combined with a defined operation (like addition or multiplication), follow specific rules until they cycle back or repeat. The interesting part comes when these abstract groups help explain the symmetries in mathematical systems or physical processes.

For instance, the group \( \mathbb{Z}_5 \) refers to integers under addition modulo 5. It forms a group because:

• Closure: The sum of any two elements still stays in the group.
• Associativity: Changing the grouping of numbers does not change their sum.
• Identity: There exists an element (0 here) that leaves numbers unchanged when added.
• Invertibility: Every element has an inverse that, when added, yields the identity.

Nontrivial Homomorphism

A homomorphism in group theory is a map between two groups that respects the group operation. A nontrivial homomorphism is an interesting scenario where this map is not simply mapping every element of the first group to the identity element of the second group.

When constructing such a homomorphism, we are looking for functions \( \phi : G \rightarrow H \) such that \( \phi(a \, b) = \phi(a) \, \phi(b) \) for all elements \( a, b \) in the group. This means the group operation in \( G \) is preserved and respected in \( H \).

• The map respects group structure, i.e., operational compatibility is a must.
• Identity elements map to identity elements of the other group.
• Nontrivial means not every element goes to the identity.

For example, in the case of \( \phi : \mathbb{Z}_5 \rightarrow \mathbb{Z}_{12} \), a nontrivial homomorphism is theoretically desired. However, due to order and divisibility constraints in our exercise, it turns out they can only map trivially, where all elements map to zero (identity) in \( \mathbb{Z}_{12} \). This stems from the fact that the condition \( 5k \equiv 0 \mod 12 \) is impossible to satisfy unless \( k = 0 \).

Remember, homomorphisms allow us to translate structure between two mathematical systems while preserving algebraic operations.