Problem 1 - A Book of Abstract Algebra

Question

In the following, let \(G\) denote an arbitrary group. Find all the normal subgroups (a) of \(S_{3}\) (b) of \(D_{4}\). Prove the following:

Step-by-Step Solution

Answer

Normal subgroups of \(S_3\) are \(\{ e \}\), \(S_3\), and \(\{ e, (12), (13), (23) \}\). Normal subgroups of \(D_4\) are \(\{ e \}\), \(D_4\), \(\{ e, r^2 \}\), and \(\{ e, r, r^2, r^3 \}\).

Step 1: Understanding Symmetric Group S3

The symmetric group \(S_3\) consists of all permutations of three objects. It has six elements: the identity permutation, three transpositions (each swapping two elements), and two 3-cycles. List these elements, which are \( \{ e, (12), (13), (23), (123), (132) \} \).

Step 2: Identifying Normal Subgroups of S3

A normal subgroup of \(S_3\) is invariant under conjugation by any element of \(S_3\). The identity subgroup \(\{ e \}\) is trivially normal. The whole group \(S_3\) is also normal. Additionally, the subgroup \(\{ e, (12), (13), (23) \}\) is normal since it includes all transpositions, and any element conjugates these within the subgroup.

Step 3: Understanding Dihedral Group D4

The dihedral group \(D_4\) represents the symmetries of a square, including rotations and reflections. It has 8 elements: four rotations and four reflections. Typically, the elements can be labeled \( \{ e, r, r^2, r^3, s, sr, sr^2, sr^3 \} \), where \( r \) represents a 90-degree rotation and \( s \) represents a reflection.

Step 4: Identifying Normal Subgroups of D4

To find normal subgroups, examine which subgroups are invariant under conjugation. The subgroups \(\{ e \}\) and \(D_4\) itself are always normal. The subgroup generated by \( r^2 \), denoted \(\{ e, r^2 \}\), is normal because conjugation by any element leaves it within the subgroup. The subgroup generated by rotations \( \{ e, r, r^2, r^3 \} \) is also normal, as reflections conjugate within it as cycles.

Key Concepts

Symmetric Group

The symmetric group, denoted as \( S_n \), is a fundamental concept in group theory. It consists of all the possible permutations of \( n \) objects. In simpler terms, it's all the different ways you can shuffle \( n \) distinct things. For example, \( S_3 \) is the symmetric group of degree 3, containing all permutations of three objects. This results in six possible permutations:

• The identity permutation \( e \), which leaves everything as is.
• Three transpositions, such as \( (12) \), \( (13) \), and \( (23) \), which swap two objects.
• Two 3-cycles \( (123) \) and \( (132) \) which move all three objects in rotation.

Understanding the symmetric group helps in identifying which of its subsets can function as normal subgroups. These normal subgroups play a crucial role in group analysis.

Dihedral Group

The dihedral group, often denoted as \( D_n \), consists of symmetries of a regular polygon with \( n \) sides, including rotations and reflections. Specifically for a square, we use \( D_4 \), which has 8 elements:

• Four rotations: the identity \( e \), a 90-degree \( r \), a 180-degree \( r^2 \), and a 270-degree rotation \( r^3 \).
• Four reflections, which can be represented as \( s, sr, sr^2, \) and \( sr^3 \).

These symmetries are known for being very geometric. By understanding the elements of \( D_4 \), one can determine normal subgroups. A subgroup in a dihedral group is normal if it remains unchanged when elements of \( D_4 \) are used to conjugate it.

Group Theory

Group theory is the study of algebraic structures known as groups. A group \( G \) consists of a set, equipped with a single operation that combines any two elements to form a third, fulfilling four conditions: closure, associativity, identity, and invertibility.
Group theory provides a foundation for many areas of mathematics and science, allowing one to analyze algebraic structures comprehensively.

• Closure: If \( a \) and \( b \) are in \( G \), then \( a \cdot b \) is also in \( G \).
• Associativity: \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \) for all \( a, b, c \) in \( G \).
• Identity: There is an element \( e \) in \( G \) such that \( e \cdot a = a \cdot e = a \) for all \( a \) in \( G \).
• Invertibility: For each \( a \) in \( G \), there is an element \( b \) in \( G \) such that \( a \cdot b = b \cdot a = e \).

Studying groups helps us explore symmetries and fundamental properties of algebraic systems.

Permutation

A permutation is a way of rearranging elements in a set. It's a bijective function from the set to itself, ensuring all elements are used exactly once. Permutations form the elements of symmetric groups, such as \( S_3 \), where each element represents a different permutation of a set of three objects.

• Identity Permutation: Leaves all elements in their original positions.
• Transpositions: Swap exactly two elements, while all others remain unchanged.
• Cycles: Involve more than two elements. A 3-cycle like \( (123) \) rearranges the first element to the position of the second, the second to the third, and the third back to the first.

Permutations are at the heart of various mathematical fields, providing a neat way to explore arrangements and transformations.

Conjugation

Conjugation is a key operation within group theory, essential for understanding normal subgroups. In a group \( G \), conjugating an element \( a \) by an element \( g \) means transforming \( a \) into \( gag^{-1} \).
Let's break down the importance of conjugation:

• Determines if a subgroup is normal: A subgroup \( H \) of \( G \) is normal if for every \( h \) in \( H \) and any \( g \) in \( G \), the conjugate \( ghg^{-1} \) is still in \( H \).
• Explores structure within groups: Conjugation showcases internal symmetry since elements linked by conjugation share properties.
• Useful in simplifying complex group operations, offering insights into equivalency and symmetry.

Understanding and applying conjugation are crucial for mastering group theory concepts, especially when identifying normal subgroups.