Problem 2 - A First Course in Abstract Algebra

Question

In Exercises I through 8 , find the order of the given factor group. \(\left(\mathrm{Z}_{4} \times \mathbb{Z}_{12}\right) /((2\rangle \times\langle 2\rangle)\)

Step-by-Step Solution

Answer

The order of the factor group is 4.

Step 1: Understand the Problem

We are given a group \( \mathbb{Z}_4 \times \mathbb{Z}_{12} \) and a normal subgroup \( \langle 2 \rangle \times \langle 2 \rangle \). We need to find the order of the factor group \( (\mathbb{Z}_4 \times \mathbb{Z}_{12}) / (\langle 2 \rangle \times \langle 2 \rangle) \).

Step 2: Determine the Order of the Original Group

The order of \( \mathbb{Z}_4 \times \mathbb{Z}_{12} \) is the product of the orders of \( \mathbb{Z}_4 \) and \( \mathbb{Z}_{12} \). Thus, \( |\mathbb{Z}_4 \times \mathbb{Z}_{12}| = 4 \times 12 = 48 \).

Step 3: Determine the Order of the Subgroup

The subgroup \( \langle 2 \rangle \times \langle 2 \rangle \) is constructed from \( \langle 2 \rangle \leq \mathbb{Z}_4 \) and \( \langle 2 \rangle \leq \mathbb{Z}_{12} \). The order of \( \langle 2 \rangle \) in \( \mathbb{Z}_4 \) is 2, and in \( \mathbb{Z}_{12} \) is 6, since \( \langle 2 \rangle = \{0, 2\} \) in \( \mathbb{Z}_4 \) and \( \{0, 2, 4, 6, 8, 10\} \) in \( \mathbb{Z}_{12} \). Thus, the order of \( \langle 2 \rangle \times \langle 2 \rangle \) is \( 2 \times 6 = 12 \).

Step 4: Use the Index Formula

The order of the factor group \((\mathbb{Z}_4 \times \mathbb{Z}_{12}) / (\langle 2 \rangle \times \langle 2 \rangle)\) is the index of the subgroup \(\langle 2 \rangle \times \langle 2 \rangle \) in \(\mathbb{Z}_4 \times \mathbb{Z}_{12}\). The index is calculated as the order of the whole group divided by the order of the subgroup: \( \frac{48}{12} = 4 \).

Key Concepts

Group Order

When dealing with group theory, understanding the concept of the "order" of a group is fundamental. The order of a group refers to the total number of elements in the group. This is an important property as it can determine other characteristics of the group. For instance:

• The order can tell us about possible subgroups.


• It can offer insights into the structure of the group.

In our original exercise, we are looking at a group defined by the Cartesian product \( \mathbb{Z}_4 \times \mathbb{Z}_{12} \). To find its order, we simply multiply the order of each component group. So, \( |\mathbb{Z}_4 \times \mathbb{Z}_{12}| = 4 \times 12 = 48 \). As seen here, the order not only calculates how many elements are present but also serves as a foundation for further computations, like finding the index of a subgroup.

Normal Subgroup

A normal subgroup is a subgroup that is invariant under group conjugation. This property is essential for forming factor (or quotient) groups. Here's why it's important:

• Allows the group to be divided into distinct cosets.


• Makes it possible to define consistent operations on cosets.

In our exercise, the subgroup \( \langle 2 \rangle \times \langle 2 \rangle \) plays the role of a normal subgroup. In the groups \( \mathbb{Z}_4 \) and \( \mathbb{Z}_{12} \), \( \langle 2 \rangle \) represents specific sets of elements:

• For \( \mathbb{Z}_4 \), \( \langle 2 \rangle = \{0, 2\} \).
• For \( \mathbb{Z}_{12} \), \( \langle 2 \rangle = \{0, 2, 4, 6, 8, 10\} \).

Each of these helps organize the group's structure so we can factorize it, maintaining its internal consistency and allowing us to explore properties of the resulting quotient group.

Index of Subgroup

In group theory, the "index" of a subgroup is the number of left cosets that can be formed from the group with that subgroup. This is directly related to the concept of the factor group or quotient group, as it represents the order of the resulting quotient group. Here's how it is determined:

• The index is calculated by dividing the total order of the group by that of the subgroup.


• This quotient is exactly the number of distinct cosets created by the subgroup in the original group.

In our exercise, we find that the subgroup \( \langle 2 \rangle \times \langle 2 \rangle \) divides the group \( \mathbb{Z}_4 \times \mathbb{Z}_{12} \) with an index of \( \frac{48}{12} = 4 \). This means there are four unique cosets, which translate to the factor group's order, telling us more about its makeup and symmetry. Understanding the index and how it relates to factor groups helps in grasping the way larger groups can be broken down into more manageable pieces.