Problem 4 - A Book of Abstract Algebra

Question

Imagine a square as a piece of paper lying on a table. The side facing you is side A. The side hidden from view is side \(B\). Every motion of the square either inter- changes the two sides (that is, side \(B\) becomes visible and side \(A\) hidden) or leaves the sides as they were. In other words, every motion \(R_{i}\) of the square brings about one of the permutations $$ \left(\begin{array}{ll} A & B \\ A & B \end{array}\right) \quad \text { or } \quad\left(\begin{array}{ll} A & B \\ B & A \end{array}\right) $$ of the sides; call it \(g\left(R_{i}\right)\). Verify that \(g: D_{4} \rightarrow S_{2}\) is a homomorphism, and give its kernel.

Step-by-Step Solution

Answer

The kernel of the homomorphism \( g: D_4 \rightarrow S_2 \) is \( \{e, r^2, \text{reflection across diagonals}\} \).

Step 1: Understanding Homomorphism

A homomorphism is a map between two algebraic structures that preserves the structures' operations. In this problem, we have a group homomorphism \( g: D_4 \rightarrow S_2 \). \( D_4 \) is the dihedral group of a square, which includes rotations and reflections, while \( S_2 \) is the symmetric group of order 2 (with two elements). We need to verify that this mapping is indeed a homomorphism by checking that it preserves the group operation.

Step 2: Analyzing Elements and Operations

The group \( D_4 \) has 8 elements that represent the symmetries of a square, including identity \( e \), rotations \( r \), \( r^2 \), \( r^3 \), and reflections. The elements of \( S_2 \) are \((e, (AB))\), representing the identity and transposition. Our permutations allow for either keeping side \( A \) as side \( A \), or interchanging \( A \) and \( B \). We need to map elements in \( D_4 \) to these permutations, i.e., group elements that either swap sides or leave them unchanged.

Step 3: Mapping Elements of \( D_4 \)

The identity \( e \), rotations \( r^2 \), and reflections across diagonals will map to \( \left( \begin{array}{cc} A & B \ A & B \end{array} \right) \) (no change in sides), while rotations \( r \), \( r^3 \), and side reflections will map to \( \left( \begin{array}{cc} A & B \ B & A \end{array} \right) \) (sides interchange). This mapping respects the operations of composition in \( D_4 \) and the resulting sides' permutations in \( S_2 \).

Step 4: Verifying Homomorphism Property

For any two elements \( x, y \in D_4 \), \( g(x \cdot y) = g(x) \cdot g(y) \) must hold. The element composition in \( D_4 \) produces a corresponding permutation in \( S_2 \) that equals the product of the independent permutations of \( x \) and \( y \). By applying the permutations as described, we observe this property holds true.

Step 5: Determining the Kernel of \( g \)

The kernel of a homomorphism is the set of elements in \( D_4 \) that map to the identity in \( S_2 \), which corresponds to \( \left( \begin{array}{cc} A & B \ A & B \end{array} \right) \). These elements that leave side \( A \) unchanged are \{e, r^2, \text{reflection across diagonals}\}. This set forms a subgroup of \( D_4 \).

Key Concepts

Dihedral Group

The dihedral group, denoted as \(D_n\), is an example of a symmetry group. It is the group of all symmetries of a regular polygon, including both rotations and reflections. For a square, which is a polygon with four sides, this group is referred to as \(D_4\).
The elements of \(D_4\) can be understood by envisioning all the actions you could perform on a square where the original arrangement can be regained by either rotating or reflecting it.
These actions include:

• Rotating 90 degrees clockwise (\(r\))
• Rotating 180 degrees (\(r^2\))
• Rotating 270 degrees clockwise (\(r^3\))
• The identity transformation where the square appears unchanged (\(e\))
• Reflections about lines through opposite vertices or midpoints of sides

The dihedral groups are vital in understanding geometric transformations and symmetries in mathematics. They are a prime example of how rotational and reflectional symmetry integrate within abstract algebra.

Symmetric Group

The symmetric group, denoted \(S_n\), is comprised of all possible permutations of a set containing \(n\) elements. For a simple set of two elements \( \{A, B\} \), the symmetric group is denoted as \(S_2\).
In this group, there are only two possible permutations:

• The identity permutation, which leaves elements unchanged: \(\left( \begin{array}{cc} A & B \ A & B \end{array} \right)\)
• A transposition that swaps the elements: \(\left( \begin{array}{cc} A & B \ B & A \end{array} \right)\)

When working with symmetry problems, \(S_2\) provides the foundational framework for understanding the basic actions elements can undergo. This introductory concept in permutations is vital when extending to larger sets in higher symmetric groups \(S_n\), which contain \(n!\) permutations. Symmetric groups play a crucial role in many areas of mathematics, including combinatorics and representation theory.

Kernel of Homomorphism

In the context of group theory, the kernel of a homomorphism is an essential concept to understand the structure of groups and their mappings. A homomorphism is a function between two groups that preserves the group action. It means that when you multiply two elements in the first group, mapping them with the homomorphism will yield the same result as multiplying their images in the second group.
The kernel of a homomorphism \(g: G \rightarrow H\) consists of all elements \(x\) in \(G\) such that \(g(x)\) is the identity in \(H\). This subset forms a group by itself, and understanding it gives insights into the nature of the homomorphism.
For example, for the homomorphism \(g: D_4 \rightarrow S_2\) described, the kernel comprises those elements of \(D_4\) that map to \(\left( \begin{array}{cc} A & B \ A & B \end{array} \right)\), the identity in \(S_2\).
In our specific case, the kernel elements \(\{e, r^2, \text{reflections across diagonals}\}\) reflect identities or symmetries that don't interchange the sides \(A\) and \(B\). Thus, analyzing the kernel offers a glimpse into symmetries unaltered by certain transformations.

Permutations

Permutations are arrangements of elements from a set into a sequence or order. In group theory, understanding permutations is crucial for comprehending how different elements are related and organized.
In the context of the symmetric group \(S_2\), permutations help us visualize the action on a pair of elements \(\{A, B\}\). There are two fundamental types of permutations in this simple group:

• The identity permutation \(\left( \begin{array}{cc} A & B \ A & B \end{array} \right)\), which leaves every element in its original position.
• The transposition permutation \(\left( \begin{array}{cc} A & B \ B & A \end{array} \right)\), which swaps the positions of the elements.

These permutations form the basis of how more complex arrangements can be constructed in larger symmetric groups. Gaining an understanding of permutations offers a foundation for mastering combinatorics and further exploring mathematical structures like rings and fields. By grasping permutations, a student can better appreciate the architecture of algebraic operations and their implications across mathematical systems.