Problem 1 - A First Course in Abstract Algebra

Question

In Exercises 1 through 8 , find the order of the given factor group. \(Z_{6} /\langle 3\rangle\)

Step-by-Step Solution

Answer

The order of \( \mathbb{Z}_6 / \langle 3 \rangle \) is 3.

Step 1: Understand the Problem

We are asked to find the order of the factor group \( \mathbb{Z}_6 / \langle 3 \rangle \). Here, \( \mathbb{Z}_6 \) is the group of integers modulo 6, and \( \langle 3 \rangle \) is the subgroup generated by 3 in \( \mathbb{Z}_6 \).

Step 2: Determine the Subgroup Generator

The subgroup \( \langle 3 \rangle \) consists of elements that are multiples of 3 in \( \mathbb{Z}_6 \). This subgroup is \{0, 3\} since in modulo 6 arithmetic, 3 added to itself stays within the set \( \{0, 1, 2, 3, 4, 5\} \).

Step 3: Calculate the Index of the Subgroup

The order of the factor group \( \mathbb{Z}_6 / \langle 3 \rangle \) is the index of \( \langle 3 \rangle \) in \( \mathbb{Z}_6 \). This index is given by the ratio \( \frac{|\mathbb{Z}_6|}{|\langle 3 \rangle|} \) where \(|\mathbb{Z}_6| = 6 \) and \(|\langle 3 \rangle| = 2\).

Step 4: Final Calculation

Compute the index: \( \frac{6}{2} = 3 \). Therefore, the order of the factor group \( \mathbb{Z}_6 / \langle 3 \rangle \) is 3.

Key Concepts

Modular Arithmetic

Modular arithmetic is often referred to as "clock math" because it deals with integers and wraps around after reaching a certain value, much like a clock resets after reaching 12. When we perform calculations in modular arithmetic, we use a number as our modulus, and all arithmetic is done "modulo" this number.
For example, in the group \( \mathbb{Z}_6 \), the numbers are \( \{0, 1, 2, 3, 4, 5\} \). Once you go beyond 5 and add 1, it wraps back to 0. This wrap-around quality makes it useful in various fields, including cryptography and computer science.

• When you see \( 7 \mod 6 \), the result is 1, because 7 divided by 6 leaves a remainder of 1.
• Similarly, \( 11 \mod 6 \) results in 5, as the remainder when 11 is divided by 6 is 5.

This method of arithmetic lets us solve problems involving cycles and repetitions efficiently.

Subgroup

In group theory, a subgroup is a set of elements that forms a group itself within a larger group, following the same operations and rules. If you're unfamiliar with group operations, think of them like a set of items already in a bigger toolbox.
The subgroup \( \langle 3 \rangle \) within \( \mathbb{Z}_6 \) is made up of all the elements you can generate by adding 3, then adding 3 again, and so forth, as much as you like. But, because we are working in modular arithmetic, these additions loop back to the beginning after reaching 6.
In our example, \( \langle 3 \rangle \) includes elements \( \{0, 3\} \), since beyond this, the additions of 3 just repeat these elements.

• Subgroups must include the identity element of the main group (here, 0), which acts like adding zero in addition.
• Subgroups must also be closed under the group operation (adding two elements from the subgroup results in another subgroup element) and include inverses for each element.

This structure ensures that subgroups retain the properties necessary to function within the larger group.

Group Theory

Group theory is a branch of mathematics that studies algebraic structures known as groups. A group is a collection of elements paired with an operation that combines any two of its elements into a third element, while following four primary conditions: closure, associativity, identity, and invertibility.
Understanding group theory helps in simplifying complex problems by examining the symmetrical structures of objects.

• Closure: For any two elements \( a \) and \( b \) in a group, the result of \( a \) combined with \( b \) (by some operation) is also in the group.
• Associativity: The group operation is associative, meaning \( (a * b) * c = a * (b * c) \).
• Identity Element: There's an element in the group, such as 0 in \( \mathbb{Z}_6 \), which, when combined with any element, leaves it unchanged.
• Inverse Element: Each element in the group has an inverse, which, when combined, returns the identity element.

Group theory applies to various mathematical fields like number theory, geometry, and more, playing a crucial role in understanding symmetries and structural properties.

Order of a Group

The order of a group is simply the number of elements it contains. For example, the group \( \mathbb{Z}_6 \) is made of six elements: \( \{0, 1, 2, 3, 4, 5\} \). But when we speak of the order of a factor group, such as \( \mathbb{Z}_6 / \langle 3 \rangle \), we're interested in the number of distinct cosets that the factor group forms.
Calculating the order of a factor group involves using the formula \( \frac{|G|}{|H|} \), where \( |G| \) is the number of elements in the original group and \( |H| \) is the number of elements in the subgroup. This tells us how many times the subgroup patterns repeat within the larger group.
In our example:

• \( |\mathbb{Z}_6| = 6 \)
• \( |\langle 3 \rangle| = 2 \)
• Thus, the order of the factor group \( \mathbb{Z}_6 / \langle 3 \rangle \) is \( \frac{6}{2} = 3 \).

Understanding group order helps in identifying the symmetry and structural features a set exhibits and is vital for decomposing groups into simpler components.