Problem 4

Question

Complete the calculation of the number of elements of order 2 in the central product of two copies of the group $D(4)$. Find how many elements of order 2 there are in the central product of two copies of the quaternion group of order eight.

Step-by-Step Solution

Verified

Answer

There are 5 elements of order 2 in the central product of two copies of $D_4$ and 3 in $Q_8$.

1Step 1: Understanding the Groups

First, identify the two groups involved in the problem. The dihedral group $ D_4 $ of order 8 has the elements $ \{e, r, r^2, r^3, s, sr, sr^2, sr^3\} $ where $ r^4 = e $ and $ s^2 = e $ with $ srs = r^{-1} $. The quaternion group $ Q_8 $ contains the elements $ \{ 1, -1, i, -i, j, -j, k, -k \} $. Both groups have elements of order 2.

2Step 2: Determine the Central Product

The central product involves taking two groups, each with a center, and factoring out elements in the center that are identified with each other. The center of $ D_4 $ is $ \{ e, r^2 \} $, and the center of $ Q_8 $ is $ \{ 1, -1 \} $. Thus, in forming the central product, $ r^2 $ and $ -1 $ will be identified with each other.

3Step 3: Elements of Order 2 in $ D_4 $ and $ Q_8 $

In $ D_4 $, elements of order 2 are $ r^2, s, sr, sr^2, sr^3 $. In $ Q_8 $, the elements of order 2 are $ -1, i^2 = -1, j^2 = -1, k^2 = -1 $, counted as distinct from $ 1 $. Understanding which elements have order 2 will aid in determining their count in the central product.

4Step 4: Calculate In Central Product for $ D_4 $

In the central product of two copies of $ D_4 $, after identifying $ r^2 $ in both groups, we need to count the ordered pairs that yield elements of order 2. These include pairs like $ (s, e), (sr, e), (sr^2, e), (sr^3, e) $ and their counterparts with the second group non-identity.

5Step 5: Calculate In Central Product for $ Q_8 $

For the central product of $ Q_8 $, after identifying $ -1 $, count distinct pairs that form order 2. Valid elements of order 2 pairs may include $ (i, -i), (j, -j), (k, -k) $, considering their behavior in the product.

6Step 6: Verify and Sum Up

Verify each element of order 2, ensuring the orders derive correctly. The elements of order 2 in each central product come from combinations of commuting elements as previously established. After verification, sum valid pairings/elements to ensure total count.

Key Concepts

Central ProductDihedral GroupQuaternion GroupOrder of an Element

Central Product

A central product combines aspects of two groups by merging them at their center, similar to stitching two pieces of fabric together at a common thread. This technique is used to create new, hybrid groups from existing ones. Each group has a center, which is the set of elements that commutate with all elements in the group. In forming the central product, elements in these centers are identified with each other.
Consider the dihedral group $ D_4 $ and the quaternion group $ Q_8 $. When forming a central product using both groups, specific elements from their centers are paired and treated as a singular element. For instance, elements like $ r^2 $ in $ D_4 $ and $ -1 $ in $ Q_8 $ become equivalent in the central product.
This process significantly affects the structure and properties of the resulting group, as it influences what order elements will have in the central product.

Dihedral Group

The dihedral group, denoted as $ D_n $, is a fundamental concept in group theory, representing symmetries of a regular polygon with $ n $ sides. For $ n = 4 $, $ D_4 $ specifically deals with the symmetries of a square. The group consists of all possible rotations and reflections that map the square onto itself.
In $ D_4 $, the elements include:

Identity element $e$
Rotations $ r, r^2, r^3 $, where each $ r$ rotation is 90 degrees
Reflections $ s, sr, sr^2, sr^3 $, corresponding to reflections across different axes

The order of an element in $ D_4 $ is defined as the smallest positive integer $ n $ where applying the element $ n $ times results in the identity element $ e $. Some elements like $ r^2 $ and reflections such as $ s $ have an order of 2, indicating they need just one repeat application to return to the original position.

Quaternion Group

The quaternion group $ Q_8 $ is an intriguing structure in mathematics, capturing properties of three-dimensional rotations using quaternions, a type of hyper-complex number. The group consists of eight elements: $ \{ 1, -1, i, -i, j, -j, k, -k \} $, and showcases unique multiplication rules.
Core elements $ i, j, k $ follow specific relations:

$ i^2 = j^2 = k^2 = -1 $
$ ij = k $, $ jk = i $, $ ki = j $
$ ji = -k $, $ kj = -i $, $ ik = -j $

In $ Q_8 $, elements such as $ i, j, k $ and their negatives have an order of 4, while $ -1 $ shows an order of 2 due to its repeated multiplication resulting in $ 1 $. The quaternion group's complex nature and multiplication rules make it a keystone in representation theory and understanding rotational symmetries.

Order of an Element

The order of an element within a group is a fundamental concept in group theory. It refers to the smallest number of times an element must be combined with itself to yield the identity element of the group.
For elements $ g $ in a group $ G $, this translates to finding the minimal $ n $ such that $ g^n = e $, where $ e $ is the identity. This evaluation provides insight into the symmetry and periodicity within a group. Elements in groups like $ D_4 $ and $ Q_8 $ show fascinating orders:

$ r^2 $ in $ D_4 $ is of order 2 because $(r^2)^2 = e$
$ -1 $ in $ Q_8 $ is also of order 2 since $(-1)^2 = 1 $

Understanding the order is crucial, especially in determining which elements contribute to the makeup of products, such as in central products, where elements of order two play a pivotal role in forming the identity through combinations of matching order.

Problem 3

Problem 5

Other exercises in this chapter

Problem 2

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Problem 3

Show that the dihedral group $D(6)$ is an internal direct product of a subgroup of order 2 with a subgroup of order 6 isomorphic to the dihedral group \(D(3)\

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Problem 5

Let $G$ be a cyclic group of order 6 generated by $x$ and $H$ be the alternating group $A(4)$, with $N$ equal to $\left\langle x^{3}\right\rangle$ a

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Problem 1

Show that the dihedral group $D(4)$ is not an internal direct product of any two of its proper subgroups.

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