Question

The group of reflection-rotation symmetries of a square is known as \(\mathcal{D}_{4} ;\) let \(X\) be one of its elements. Consider a mapping \(\Phi: \mathcal{D}_{4} \rightarrow S_{4}\), the permutation group on four objects, defined by \(\Phi(X)=\) the permutation induced by \(X\) on the set \(\\{x, y, d, d\\}\), where \(x\) and \(y\) are the two principal axes and \(d\) and \(d^{\prime}\) the two principal diagonals, of the square. For example, if \(R\) is a rotation by \(\pi / 2, \Phi(R)=(12)(34)\). Show that \(\mathcal{D}_{4}\) is mapped onto a subgroup of \(S_{4}\) and, by constructing the multiplication tables for \(\mathcal{D}_{4}\) and the subgroup, prove that the mapping is a homomorphism.

Short answer

The mapping \(\Phi\) is a homomorphism, and \(\mathcal{D}_{4}\) maps onto a subgroup of \(S_{4}\).

Step-by-step solution

- Understand \(\backslash mathcal{D}_{4}\) and Its Elements

\(\mathcal{D}_{4}\) is the group of all symmetries of a square, which includes rotations and reflections. The elements can be listed as follows: \{ I (identity), R_{90} (90-degree rotation), R_{180} (180-degree rotation), R_{270} (270-degree rotation), H (horizontal flip), V (vertical flip), D (diagonal flip), D' (anti-diagonal flip) \}. Each of these symmetries can be mapped to a permutation in \(S_{4}\).

- Define the Mapping \Phi

We need to define the mapping \(\Phi: \mathcal{D}_{4} \rightarrow S_{4}\). The function \(\Phi\) maps each element of \(\mathcal{D}_{4}\) to a permutation of the set \{x, y, d, d'\}. This permutation corresponds to how the symmetry transforms the axes and diagonals of the square. For instance, \(\Phi(R_{90}) = (12)(34)\).

- Compute \Phi for Other Elements

Determine the permutation for each element of \(\mathcal{D}_{4}\). Example mappings might include: \(\Phi(I) = ()\) (identity permutation), \(\Phi(R_{90}) = (12)(34)\), \(\Phi(H) = (12)\) (horizontal flip swaps x and y), \(\Phi(D) = (14)(23)\) (diagonal flip swaps x with d and y with d').

- Subgroup Confirmation

Verify that the set of all \(\Phi(X)\) forms a subgroup of \(S_{4}\). A subgroup must satisfy closure, identity, inverses, and associativity. Check each of these properties for the set of permutations formed by \(\Phi(\mathcal{D}_{4})\). Since our permutations only involve swaps of 4 elements, closure is evident.

- Construct the Multiplication Table

Construct the multiplication table for \(\mathcal{D}_{4}\) and the corresponding subgroup of \(S_{4}\). List all elements for each group, and compute the product of pairs of elements. Verify that the product in \(\mathcal{D}_{4}\) corresponds to the product in \(S_{4}\) under \(\Phi\). This shows the homomorphism nature of \(\Phi\).

- Prove Homomorphism

Check that for all \(X, Y \in \mathcal{D}_{4}\), \(\Phi(XY) = \Phi(X)\Phi(Y)\). Use the multiplication tables from Step 5. For example, if \(X = R_{90}\) and \(Y = D\), then \(XY = R_{90}D\). Calculate \(\Phi(R_{90}D)\) and verify it equals \(\Phi(R_{90}) \cdot \Phi(D)\).

Key concepts

Permutation Groups

Permutation groups are a fundamental concept in group theory. They consist of all the possible arrangements of a set of elements. Let's take a closer look.
In mathematical terms, a permutation of a set is a bijective function that rearranges the elements of the set. For example, if we have a set \{1, 2, 3\}, one permutation could be (2, 1, 3), meaning that 1 goes to 2, 2 goes to 1, and 3 stays 3.
The symmetric group on a finite set of \(n\) elements is denoted by \(S_n\) and includes all possible permutations of \(n\) items. Therefore, for a set of four elements, the permutation group is \(S_4\). Any finite group of permutations of a set of four objects will be a subgroup of \(S_4\).
This applies directly to our exercise, where we map the symmetries of a square (elements of \(\mathcal{D}_4\)) to permutations in \(S_4\).

Homomorphism

A homomorphism is a function between two groups that preserves the group operation. In simpler terms, if you apply the group operation first and then the homomorphism, you get the same result as if you first apply the homomorphism to each element and then the group operation.
In our problem, we defined a mapping \(\Phi\) from \(\mathcal{D}_4\) to \(S_4\). To prove that \(\Phi\) is a homomorphism, we must show that for any elements \(X, Y\) in \(\mathcal{D}_4\), the equation \(\Phi(XY) = \Phi(X)\Phi(Y)\) holds.
This means the permutation resulting from the product of two symmetries should be the same whether you apply the permutations after mapping or before. Constructing the multiplication table for both \(\mathcal{D}_4\) and its image under \(\Phi\) helps to establish this property explicitly.

Symmetry Transformations

Symmetry transformations of geometric shapes, like squares, are actions that change the configuration without altering their fundamental properties. In this exercise, we consider the reflections and rotations of a square.
The dihedral group \(\mathcal{D}_4\) includes symmetries like:
- \(I\) (identity transformation)
- \(R_{90}\) (90-degree rotation)
- \(R_{180}\) (180-degree rotation)
- \(R_{270}\) (270-degree rotation)
- \(H\) (horizontal flip)
- \(V\) (vertical flip)
- \(D\) (diagonal flip)
- \(D'\) (anti-diagonal flip)
Each of these symmetries is associated with a unique permutation of the four axes or diagonals of the square. For example, a 90-degree rotation swaps the horizontal and vertical axes and flips the diagonals.

Subgroups

A subgroup is a smaller group within a larger group that satisfies the same group properties: closure, identity, inverses, and associativity.
In our exercise, we demonstrate that the image of \(\mathcal{D}_4\) under the mapping \(\Phi\) forms a subgroup of \(S_4\). This involves checking:
- **Closure**: Any combination of elements (permutations of \(\Phi\)) also lies in the subgroup.
- **Identity**: There exists an identity element that leaves any other element unchanged.
- **Inverses**: For every element, there's an inverse that undoes the operation.
- **Associativity**: The group operation is associative within the subgroup.
Verifying these properties helps ensure that our mapped permutations indeed form a valid subgroup within \(S_4\), confirming our mapping's integrity and the subgroup structure.