Problem 6 - A Book of Abstract Algebra

Question

In each of the following, \(G\) is a group and \(H\) is a normal subgroup of \(G\). List the elements of \(G / H\) and then write the table of \(G / H\). Example \(G=\mathbb{Z}_{6} \quad\) and \(\quad H=\\{0,3\\}\) The elements of \(G / H\) are the three cosets \(H=H+0=\\{0,3\\}, H+1=\\{1,4\\}\), and \(H+2=\\{2,5\\} .\) (Note that \(H+3\) is the same as \(H+0, H+4\) is the same as \(H+1\), and \(H+5\) is the same as \(H+2\).) The table of \(G / H\) is $$ \begin{array}{c|ccc} \+ & H & H+1 & H+2 \\ \hline H & H & H+1 & H+2 \\ H+1 & H+1 & H+2 & H \\ H+2 & H+2 & H & H+1 \end{array} $$ \(G=P_{3}, H=\\{\emptyset,\\{1\\}\\} .\left(P_{3}\right.\) is the group of subsets of \(\left.\\{1,2,3\\} .\right)\)

Step-by-Step Solution

Answer

The elements of \( G/H \) are \( H, H+\{2\}, H+\{3\}, H+\{2,3\} \). The table shows the group's structure, revealing four distinct elements forming a Klein four-group.

Step 1: Identify the Elements of G and H

In this case, we have the group \( G = P_3 \), which is the power set of the set \( \{1, 2, 3\} \). This means \( G \) is composed of all subsets of \( \{1, 2, 3\} \), which are \( \emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\} \). The subgroup \( H = \{ \emptyset, \{1\} \} \).

Step 2: Form the Cosets of G / H

To find cosets of \( G / H \), we compute \( H + A \) where \( A \) is each element of \( G \). The addition here is symmetric difference. The distinct cosets we find are:1. \( H + \emptyset = H = \{\emptyset, \{1\}\} \)2. \( H + \{2\} = \{\{2\}, \{1, 2\}\} \)3. \( H + \{3\} = \{\{3\}, \{1, 3\}\} \)4. \( H + \{2, 3\} = \{\{2, 3\}, \{1, 2, 3\}\} \)

Step 3: Verify and List Distinct Cosets

Check if any other element in \( G \) would produce a new coset:- \( \{1, 2\}, \{1, 3\}, \{1, 2, 3\} \) will produce the same cosets as above because of overlaps with elements. Therefore, our distinct cosets for \( G / H \) are:- \( H \)- \( H + \{2\} \)- \( H + \{3\} \)- \( H + \{2, 3\} \)

Step 4: Construct the Cayley Table for G / H

Construct the table using the operation of symmetric difference for the cosets:\[\begin{array}{c|cccc}+ & H & H+\{2\} & H+\{3\} & H+\{2,3\} \\hlineH & H & H+\{2\} & H+\{3\} & H+\{2,3\} \H+\{2\} & H+\{2\} & H & H+\{2,3\} & H+\{3\} \H+\{3\} & H+\{3\} & H+\{2,3\} & H & H+\{2\} \H+\{2,3\} & H+\{2,3\} & H+\{3\} & H+\{2\} & H \\end{array}\]

Step 5: Interpret the Result

The Cayley table reveals the structure of the quotient group \( G/H \), which possesses four distinct elements, confirming that \( G/H \) is isomorphic to \( \, \mathbb{Z}_2 \times \mathbb{Z}_2 \).

Key Concepts

Normal Subgroup

In group theory, a subgroup is referred to as **normal** if it satisfies a specific condition. This condition states that for every element in the group, when you conjugate it with an element of the subgroup, it remains within the subgroup. Mathematically, this means that for any element \( g \) in the group \( G \) and any element \( h \) in the subgroup \( H \):

• \( gHg^{-1} = H \)

This elegant symmetry allows us to form a new group structure, known as the quotient group.
Think of a normal subgroup as having a harmonious relationship with the entire group, where its elements mesh seamlessly within the operation of the group.
For instance, in our example, \( H = \{ \emptyset, \{1\} \} \) is a normal subgroup of \( G = P_3 \), providing the foundation for further exploration of cosets and quotient groups.

Cosets

Cosets are a way to partition a group into non-overlapping subsets. When you have a subgroup \( H \) within a group \( G \), you can create cosets by "adding" (in a suitable group operation sense like multiplication or, in this case, symmetric difference) elements from \( G \) to elements of \( H \).

• For a left coset, \( gH = \{ gh : h \in H \} \)

Think of cosets as defining a certain shift across the group's structure while still staying connected to the subgroup.
In our exercise, we found cosets through the operation of symmetric difference:

• \( H = \{ \emptyset, \{1\} \} \)
• \( H + \{2\} = \{ \{2\}, \{1, 2\} \} \)
• ... and so on.

Each distinct coset is a building block in the formation of the quotient group, demonstrating group symmetry in a new form.

Quotient Group

A **quotient group** is what you get when you divide a group by one of its normal subgroups. Symbolically, it is represented as \( G / H \), where \( G \) is the group and \( H \) is its normal subgroup.
Forming a quotient group involves collecting all possible cosets of \( H \) in \( G \) and using them as the elements of a new group.
This new group inherits its operation from \( G \), but its elements are these cosets instead of single group elements. Each coset acts almost like a single, indivisible entity in this new structure.
In our example, the quotient group \( G/H \) consists of the cosets \( H \), \( H + \{2\} \), and so on. These cosets are the fundamental units of the quotient group.

Cayley Table

A **Cayley table** is a valuable tool for visualizing the structure of a group. It lists all the elements of the group both horizontally and vertically, and shows the result of their combination according to the group operation.
In our example, the group operation is symmetric difference. The table demonstrates how cosets within the quotient group \( G/H \) interact with each other.

• The entry at row \( A \) and column \( B \) shows the result of the group operation on \( A \) and \( B \).

By studying the Cayley table, a pattern or symmetry can be observed. This pattern reflects the algebraic structure and behavior of the quotient group, providing insight into its properties.
For \( G = P_3 \) and \( H = \{ \emptyset, \{1\} \} \), the table reveals a structure akin to \( \mathbb{Z}_2 \times \mathbb{Z}_2 \), a well-understood group in abstract algebra.