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Chapter 15: Problem 4

In its realization as permutations, the dihedral group \(\mathcal{D}_{3}\) is equal to \(S_{3}\). Can you give a geometric explanation why? Why isn't \(\mathcal{D}_{4}\) equal to \(S_{4}\) ?

Short Answer

Expert verified
D_3 equals S_3 because their symmetries match; D_4 isn't S_4 as it lacks enough permutations.

Step by step solution

01

Understanding D_3 as Permutations

The dihedral group D_3 represents the symmetries of an equilateral triangle, including rotations and reflections. It has 3 rotations (06, 1206, 2406) and 3 reflections, making a total of 6 transformations. These symmetries can permute the 3 vertices of the triangle, corresponding directly to some of the permutations in the symmetric group S_3, which also contains 6 elements, matching the order of D_3.
02

Rotations and Reflections in S_3

Every symmetry in D_3 can be seen as a specific permutation of the triangle's vertices. For instance, a 1206 rotation moves vertex 1 to position 2, vertex 2 to position 3, and vertex 3 to position 1 鈥� a permutation cycle (123). Similar operations can be mapped to the other elements of S_3, proving D_3 is equivalent to S_3 as they share the same operations.
03

Understanding D_4 as Permutations

D_4 denotes the dihedral group of a square, comprising 8 symmetries: 4 rotations (06, 906, 1806, 2706) and 4 reflections. Unlike the equilateral triangle, a square has 4 vertices that can be permuted.
04

Comparing D_4 and S_4

The symmetric group S_4 includes all possible permutations of 4 elements, totaling 24 permutations, while D_4 only contains the 8 symmetries of a square. Therefore, D_4 does not have enough elements to account for all the permutations in S_4, indicating D_4 4 S_4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Symmetric Group
A symmetric group, denoted as \(S_n\), is a mathematical structure that relates to permutations. It consists of all possible permutations, or arrangements, of \(n\) objects. For example, if we have 3 objects, say \(a, b, \) and \(c\), the symmetric group \(S_3\) includes all permutations of these items, which totals to \(3! = 6\) permutations.
  • In the context of geometry, the symmetric group helps illustrate how different transformations can rearrange the vertices of figures like triangles or squares.
  • Symmetric groups are important in the study of both algebra and geometry, providing a framework for understanding how objects can be symmetrically manipulated.
Through understanding symmetric groups, we can decode the complex symmetries of shapes and their transformations.
Permutations
Permutations are rearrangements of a set of objects. For example, rearranging the three vertices of a triangle in all possible ways results in 6 arrangements or permutations. This corresponds to the 6 elements found in the symmetric group \(S_3\). These permutations can be understood as specific operations that reposition the vertices, allowing us to explore different symmetry operations in geometric figures.
  • Each permutation can be represented by a cycle notation. For instance, the cycle (123) represents the rotation of vertices so that vertex 1 moves to position 2, vertex 2 to position 3, and vertex 3 returns to position 1.
  • Understanding permutations helps in exploring the symmetries of geometric shapes, such as equilateral triangles and squares.
Permutations form the basis for how we comprehend the arrangement possibilities of any set or group of objects.
Geometric Symmetry
Geometric symmetry involves transformations that leave a geometrical object unchanged in size and shape, although possibly repositioned. In simpler terms, it鈥檚 the ability to move a shape in various ways without altering its overall appearance.
  • Types of geometric symmetry include rotations, reflections, and translations.
  • These operations form the essence of symmetry groups like dihedral groups for polygons.
By examining geometric symmetry, we identify transformations, such as those found in an equilateral triangle's dihedral group \(\mathcal{D}_3\), which relate directly to permutations of the shape. Understanding these transformations helps decode the inherent balance and consistency within geometric figures.
Equilateral Triangle
An equilateral triangle is a triangle where all sides are of equal length and all internal angles are 60 degrees. This uniformity gives it a unique set of six symmetries, forming what is known as the dihedral group \(\mathcal{D}_3\). These symmetries include three rotations and three reflections, aligning perfectly with the permutations in \(S_3\).
  • Rotations of 0, 120, and 240 degrees reposition the triangle's vertices, effectively acting as permutations of the vertices.
  • Reflections in the axes of symmetry also rearrange the positions of the vertices.
The equilateral triangle's symmetries provide a perfect practical example of how geometric transformations can be understood through symmetric groups and permutations.
Square Symmetries
Square symmetries are transformations that can be performed on a square, comprising a total of 8 distinct symmetries. These include 4 rotations (0, 90, 180, and 270 degrees) and 4 reflections across axes of symmetry.
  • The dihedral group \(\mathcal{D}_4\) consists of these symmetries, demonstrating how a square can be repositioned without altering its appearance.
  • However, the symmetric group \(S_4\), relating to permutations of 4 elements, has 24 elements. Thus, \(\mathcal{D}_4\) with only 8 elements, isn't equivalent to \(S_4\).
This discrepancy highlights how different geometric shapes, like squares, relate to symmetric and dihedral groups differently based on their inherent symmetries and transformations.

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Most popular questions from this chapter

Consider \(\mathbb{Z}_{10}\) and the subsets of \(\mathbb{Z}_{10},\\{0,1,2,3,4\\}\) and \\{5,6,7,8,9\\} . Why is the operation induced on these subsets by modulo 10 addition not well defined?

Prove by induction that if \(r \geq 1\) and each \(t_{i},\) is a transposition, then \(\left(t_{1} \circ t_{2} \circ \cdots \circ t_{r}\right)^{-1}=t_{r} \circ \cdots \circ t_{2} \circ t_{1}\)

Which of the following groups are cyclic? Explain. (a) \([\mathbb{Q} ;+]\) (b) \(\left[\mathbb{R}^{+} ; \cdot\right]\) (c) \([6 \mathbb{Z} ;+]\) where \(6 \mathbb{Z}=\\{6 n \mid n \in \mathbb{Z}\\}\) (d) \(\mathbb{Z} \times \mathbb{Z}\) (e) \(\mathbb{Z}_{2} \times \mathbb{Z}_{3} \times \mathbb{Z}_{25}\)

Prove that if \(G\) is an abelian group, then \(q(x)=x^{2}\) defines a homomorphism from \(G\) into \(G\). Is \(q\) ever an isomorphism?

Prove that if \(\theta: G \rightarrow G^{\prime}\) is a homomorphism, and \(H \triangleleft G,\) then \(\theta(H) \triangleleft \theta(G)\). Is it also true that \(\theta(H) \triangleleft G^{\prime} ?\)
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