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Chapter 15: Problem 5

(a) Complete the list of elements of \(\mathcal{D}_{4}\) and write out a table for the group in its realization as symmetries. (b) List the subgroups of \(\mathcal{D}_{4}\) in a lattice diagram. Are they all cyclic? To what simpler groups are the subgroups of \(\mathcal{D}_{4}\) isomorphic?

Short Answer

Expert verified
(a) \\(\mathcal{D}_{4}\u00a7f\\) includes rotations and reflections. (b) Subgroups are not all cyclic; some are isomorphic to \\(\mathbb{Z}_2\u00a7f\\) and \\(\mathbb{Z}_4\u00a7f\\).

Step by step solution

01

Understanding the Symmetries of a Square

The group \(\mathcal{D}_{4}\u00a7f\), or the dihedral group of order 8, represents the symmetries of a square. This includes rotations and reflections. The possible rotations are 0, 90, 180, and 270 degrees. Reflections can be made across the vertical axis, horizontal axis, and the two diagonals, giving a total of 8 elements.
02

Listing Elements of \\(\\mathcal{D}_{4}\u00a7f\\)

Elements are: \(e\u00a7f\) (identity), \(r\u00a7f\) (rotation 90 degrees), \(r^{2}\u00a7f\) (rotation 180 degrees), \(r^{3}\u00a7f\) (rotation 270 degrees), \(s_{v}\u00a7f\) (vertical reflection), \(s_{h}\u00a7f\) (horizontal reflection), \(d_{1}\u00a7f\) (reflection across one diagonal), and \(d_{2}\u00a7f\) (reflection across the other diagonal).
03

Creating the Group Table

Construct a Cayley table for \(\mathcal{D}_{4}\u00a7f\) by calculating the result of applying each pair of operations. Use a systematic approach to combine rotations with reflections, ensuring to list results in a table format that shows the outcome when applying these operations sequentially.
04

Identifying Subgroups

To find subgroups, consider sets of elements that themselves form a group under the same operation. Obvious subgroups include rotations \(\{e, r, r^2, r^3\u00a7f\}\), reflections such as \(\{e, s_{v}\u00a7f\}\) or \(\{e, d_{1}\u00a7f\}\), and others found by exploring combinations.
05

Structuring a Lattice Diagram

The lattice diagram should display all identified subgroups, showing inclusions. \(\mathcal{D}_{4}\u00a7f\) is at the top, \(\{e\}\u00a7f\) at the bottom, and subgroups in between reflecting sets previously identified.
06

Analyzing Cyclic Properties

Check if each subgroup is cyclic; a group is cyclic if all elements can be generated by repeated application of a single element. This involves checking subgroups like \(\{e, r, r^2, r^3\u00a7f\}\) for cyclic behavior.
07

Isomorphic Subgroups

Identify simpler groups to which subgroups are isomorphic. The subgroup of rotations \(\{e, r, r^2, r^3\u00a7f\}\) is isomorphic to \(\mathbb{Z}_4\u00a7f\), and reflection subgroups like \(\{e, s_v\u00a7f\}\) are isomorphic to \(\mathbb{Z}_2\u00a7f\).

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Key Concepts

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

Symmetries of a Square
When we talk about the symmetries of a square, we're referring to all the ways you can move the square so that it looks exactly the same as it did before. These moves include both rotations and reflections.
  • **Rotations:** The square can be rotated about its center by 0, 90, 180, or 270 degrees. Each rotation results in the square looking the same as its original position.
  • **Reflections:** The square can be reflected across various axes: the vertical axis, the horizontal axis, and two diagonals (which are imaginary lines running from corner to corner of the square).
Together, these symmetries form a group, called the dihedral group of order 8, or \(\mathcal{D}_{4}\). This group includes 8 unique operations, each manipulating the square's appearance but maintaining its symmetry.
Cayley Table
The Cayley table is a handy tool for understanding how different symmetries work together. It's similar to a multiplication table, but instead of numbers, we're dealing with symmetries like rotations and reflections.
To create this table for \(\mathcal{D}_{4}\):
  • List all elements: the identity element (doing nothing), rotations \(r, r^2, r^3\), and reflections \(s_v, s_h, d_1, d_2\).
  • Calculate the result of applying any two operations in sequence. For example, applying two 90-degree rotations results in a 180-degree rotation.
  • Fill the table systematically, ensuring each entry corresponds to the combined result of two symmetries. This helps visualize how all operations in \(\mathcal{D}_{4}\) interact with each other.
The Cayley table provides a comprehensive view of the symmetry group's structure.
Subgroups
Subgroups are smaller groups that can be formed from the main group, having their own elements and operations. With \(\mathcal{D}_{4}\), we explore various sets of symmetries that form subgroups.
  • **Rotational Subgroup:** This includes only rotations: \(\{e, r, r^2, r^3\}\). This is a group because performing any sequence of these rotations also produces a rotation in the set.
  • **Reflectional Subgroups:** These include, for example, \(\{e, s_v\}\), where you have just the identity and vertical reflection.
  • **Combining Operations:** There are more subgroups discovered by combining other symmetries to form valid groups adhering to group properties.
Identifying subgroups is essential for understanding the internal structure of more complicated groups like \(\mathcal{D}_{4}\).
Cyclic Groups
Cyclic groups are a specific type of group where every element can be generated by repeatedly applying a single element. Not all groups are cyclic, but some subgroups of \(\mathcal{D}_{4}\) are.
  • **Example of a Cyclic Subgroup:** The subgroup of rotations \(\{e, r, r^2, r^3\}\) is cyclic. Here, any rotation can be reached by applying \(r\) multiple times.
  • **Non-Cyclic Subgroups:** Reflection groups like \(\{e, s_v\}\) are not cyclic because you need more than one element to describe all operations.
Identifying cyclic properties helps in simplifying the group's analysis and can provide insights into subgroup structures and classifications. Cyclic groups often correspond to simpler, more well-known mathematical structures, such as integer groups modulo some number, denoted as \(\mathbb{Z}_n\).

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Most popular questions from this chapter

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 \(\theta(H) \triangleleft G^{\prime} ?\)

Consider the linear code defined by the generator matrix $$ G=\left(\begin{array}{llll} 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 \end{array}\right) $$ (a) What size blocks does this code encode and what is the length of the code words? (b) What are the code words for this code? (c) With this code, can you detect single bit errors? Can you correct all, some, or no single bit errors?

Define the two functions \(\alpha: \mathbb{Z}_{2}^{3} \rightarrow \mathbb{Z}_{2}^{4}\) and \(\beta: \mathbb{Z}_{2}^{4} \rightarrow \mathbb{Z}_{2}\) by \(\alpha\left(a_{1}, a_{2}, a_{3}\right)=\) \(\left(a_{1}, a_{2}, a_{3}, a_{1}+_{2} a_{2}+_{2} a_{3}\right),\) and \(\beta\left(b_{1}, b_{2}, b_{3}, b_{4}\right)=b_{1}+b_{2}+b_{3}+b_{4}\) Describe the function \(\beta \circ \alpha\). Is it a homomorphism?

For each group and element, determine the order of the cyclic subgroup generated by the element: (a) \(\mathbb{Z}_{25}, 15\) (b) \(\mathbb{Z}_{4} \times \mathbb{Z}_{9},(2,6)\) (apply Exercise 8) (c) \(\mathbb{Z}_{64}, 2\)

Assume that \(G\) is a group, \(H \leq G,\) and \(a, b \in G\). Prove that \(a * H=b * H\) if and only if \(b^{-1} * a \in H\).
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