Question

In the quaternion group \(Q\) the elements form the set $$ \\{1,-1, i,-i, j,-j, k,-k\\} $$ with \(i^{2}=j^{2}=k^{2}=-1, i j=k\) and its cyclic permutations, and \(j i=-k\) and its cyclic permutations. Find the proper subgroups of \(Q\) and the corresponding cosets. Show that the subgroup of order 2 is a normal subgroup, but that the other subgroups are not. Show that \(Q\) cannot be isomorphic to the group \(4 m m\) \(\left(C_{4 c}\right)\) considered in exercise \(24.11 .\)

Short answer

The proper subgroups are \{1, -1\}, \{1, -1, i, -i\}, \{1, -1, j, -j\}, \{1, -1, k, -k\}. Only \{1, -1\} is a normal subgroup. Group \( Q \) is not isomorphic to \( C_{4c} \).

Step-by-step solution

- Identify Elements and Group Operations

The elements of the quaternion group \( Q \) are \( \{ 1, -1, i, -i, j, -j, k, -k \} \). The multiplication rules include \( i^2 = j^2 = k^2 = -1 \), \( ij = k \), \( jk = i \), \( ki = j \), and \( ji = -k \), with their cyclic permutations. Additionally, \(k j = -i \) and \( i k = -j \).

- Find Proper Subgroups

List all possible subgroups of \( Q \). Subgroups by definition must include the identity element (1) and be closed under the group operation. Possible subgroups are: \[ \{1, -1\} \],\[ \{1, -1, i, -i\} \],\[ \{1, -1, j, -j\} \],\[ \{1, -1, k, -k\} \].

- Identify and Confirm Cosets

To find the cosets of each subgroup, calculate the left cosets for each. For subgroup \( \{1, -1\} \): \( i \{1, -1\} = \{i, -i\}, j \{1, -1\} = \{j, -j\}, k \{1, -1\} = \{k, -k\} \). Similarly, calculate for all subgroups.

- Prove Normality

For subgroup \( \{1, -1\} \), check for normality by showing all left cosets are equal to right cosets (i.e., cosets do not depend on the side of multiplication): For instance, \( i \{1, -1\} = \{i, -i\} \) and \( \{1, -1\} i = \{i, -i\} \). Therefore, \( \{1, -1\} \) is a normal subgroup.

- Show Non-Normality of Other Subgroups

For other subgroups, such as \(\{1, -1, i, -i\} \), compare left and right cosets. For example, the coset calculations give different results, hence the subgroups are not normal.

- Show Non-Isomorphism to Group \(4mm (C_{4c})\)

Consider the properties of group \( C_{4c} \) from the earlier exercise. Group \( C_{4c} \) has different order and structure compared to quaternion group \( Q \). Hence, no isomorphism exists between them.

Key concepts

Group Theory

Group theory is the study of algebraic structures known as groups. A group is a set equipped with a binary operation that combines any two elements to form a third element. This operation must obey four properties: closure, associativity, identity, and invertibility.

A group is called abelian if the group operation is commutative. This means that, for any elements \(a\) and \(b\) in the group, the equation \(ab = ba\) holds. Non-abelian groups, like the quaternion group \(Q\), do not satisfy this property.

In the quaternion group \(Q\), we have the elements \(\{1, -1, i, -i, j, -j, k, -k\}\). The group operation is quaternion multiplication, governed by specific rules like \(i^2 = j^2 = k^2 = -1\) and \(ij = k\), with various cyclic permutations.

Understanding these foundational properties helps us explore concepts such as subgroups, cosets, and isomorphism effectively.

Subgroups

A subgroup is a subset of a group that is itself a group with the same operation. In the context of the quaternion group \(Q\), we need to identify all subsets that satisfy group properties.

Subgroups of \(Q\) have to include the identity element (1) and be closed under quaternion multiplication. Here are some examples of subgroups:

• \(\{1, -1\}\)
• \(\{1, -1, i, -i\}\)
• \(\{1, -1, j, -j\}\)
• \(\{1, -1, k, -k\}\)

Each subgroup holds its own unique properties, but only \(\{1, -1\}\) will turn out to be a normal subgroup as demonstrated through coset calculations.

Cosets

Cosets help us understand the division of a group into equal parts or 'cosets' formed by multiplying all elements of a subgroup by a fixed element from the group.

Left cosets of a subgroup \(H\) in a group \(G\) are found by multiplying \(H\) by each element from the left, whereas right cosets multiply from the right. For example, given subgroup \(\{1, -1\}\) in the quaternion group, we compute:

• Left cosets: \( i\{1, -1\} = \{i, -i\}\), \( j\{1, -1\} = \{j, -j\}\), etc.
• Right cosets are computed similarly and for \(\{1, -1\}\) they match left cosets, hence showing it is normal.

By comparing left and right cosets for other subgroups such as \(\{1, -1, i, -i\}\), we find differences, proving these subgroups are not normal.

Isomorphism

Isomorphism in group theory is a bijective homomorphism between two groups. If such a mapping exists, the groups are said to be isomorphic, meaning they have the same structure.

To check if the quaternion group \(Q\) is isomorphic to group \(C_{4c}\), we compare their elements and operations. A careful look reveals that the order and algebraic structures of these groups differ fundamentally. For instance, the quaternion group \(Q\) has 8 elements with specific multiplication rules, while \(C_{4c}\) has different properties as discussed in a different exercise chapter.

As no one-to-one correspondence fulfilling group operations exists between these groups, they are not isomorphic.