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Magazine Chapter 4: Problem 9
Es sei \(Q=\\{E,-E, I,-I, J,-J, K,-K\\}\) die Quaternionengruppe (siehe Beispiel 2.1). Bestimmen Sie alle Untergruppen und alle Normalteiler von \(Q\).
Short Answer
Expert verified
The subgroups are \( \{E\}, \{E, -E\}, \{E, I\}, \{E, -I\}, \{E, J\}, \{E, -J\}, \{E, K\}, \{E, -K\}, Q\). The normal subgroups are \( \{E\}, Q, \{E, I, -E, -I\}, \{E, J, -E, -J\}, \{E, K, -E, -K\} \).
Step by step solution
01
Understanding the Quaternion Group
The quaternion group, denoted as \( Q \), has 8 elements: \( \{ E, -E, I, -I, J, -J, K, -K \} \). Each element has specific multiplication rules. Our goal is to find all subgroups and all normal subgroups.
02
Determine Subgroups of Order 1 and 2
For any group, the trivial subgroup \( \{E\} \) is always a valid subgroup. Additionally, any element along with the identity \( E \) forms a subgroup of order 2. Therefore, the subgroups of order 2 are \( \{ E, -E \} \), \( \{ E, I \} \), \( \{ E, -I \} \), \( \{ E, J \} \), \( \{ E, -J \} \), \( \{ E, K \} \), and \( \{ E, -K \} \).
03
Check for Larger Subgroups
Subgroups of a group are of order equal to divisors of the group's order. Since \( Q \) has order 8, potential divisors are 1, 2, 4, and 8. We have identified order 1 and 2 subgroups. Next, we check for order 4 subgroups, which will include elements that square to the identity. Using the relations between elements, we find \( \{E, I, -E, -I \} \), \( \{E, J, -E, -J \} \), and \( \{E, K, -E, -K \} \) as valid subgroups.
04
Identify the Quaternion Group as a Subgroup
Additionally, the entire set \( Q \) itself is also a subgroup of order 8, as any group is a trivial subgroup of itself.
05
Determine Normal Subgroups
A subgroup is normal if it is invariant under conjugation by any element of the group. The subgroups of order 1 \( \{E\} \) and order 8 \( Q \) are trivial normal subgroups by definition. Each order 4 subgroup identified is normal because they commute during conjugation within the quaternion algebra (e.g., \( i \cdot j \cdot i^{-1} = j \) for elements \( i \) and \( j \)). These are \( \{E, I, -E, -I \} \), \( \{E, J, -E, -J \} \), and \( \{E, K, -E, -K \} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Subgroups
In group theory, subgroups are smaller groups formed from a larger group by using some of its elements and maintaining the group's operational structure. When dealing with the Quaternion Group, which is denoted by \( Q \) and contains 8 elements \( \{E, -E, I, -I, J, -J, K, -K\} \), we start by identifying its subgroups. Subgroups can have orders that are divisors of the group's total number of elements. For \( Q \), which has 8 elements, potential subgroup orders are 1, 2, 4, and 8.
- The trivial subgroup, \( \{E\} \), contains only the identity element and is always a subgroup of any group.
- Subgroups of order 2 include pairs combining identity with one other element, such as \( \{E, -E\} \) and \( \{E, I\} \).
- Order 4 subgroups are constructed by identifying elements that mutually satisfy the subgroup criteria, like \( \{E, I, -E, -I\} \).
- Lastly, the whole group \( Q \) itself is a subgroup of order 8.
Normal Subgroups
Normal subgroups are a special kind of subgroup that remain invariant under conjugation by any element of the parent group. In simpler terms, for a subgroup \( N \) to be normal in group \( G \), it must satisfy the condition that for every element \( g \) in \( G \) and every element \( n \) in \( N \), the product \( gng^{-1} \) is still in \( N \).For the quaternion group \( Q \):
- The most obvious normal subgroups are the trivial ones, \( \{E\} \) and the entire group \( Q \) itself.
- All order 4 subgroups in \( Q \) are normal because they commute with other elements during multiplication, meaning conjugation does not change them.
Group Order
The order of a group is the number of elements it contains. For the quaternion group \( Q \), the order is 8 since it contains exactly 8 elements: \( \{ E, -E, I, -I, J, -J, K, -K \} \).Group order is significant because it influences the possible sizes of subgroups. By Lagrange's Theorem, the order of any subgroup must divide the order of the entire group.
- This means in our group \( Q \), only subgroups of sizes 1, 2, 4, and 8 are possible.
- The relation of group order with subgroups helps us predict and verify the correctness of subgroups discovered.
Multiplication Rules
The multiplication rules of a group define how the elements of the group interact with each other through the group鈥檚 operation, typically denoted by multiplication or another binary operation.In the quaternion group \( Q \), these rules are complex and extend beyond elementary arithmetic, supporting interactions like rotations in three-dimensional space. Here are the basic rules:
- The identity element \( E \) acts like number 1 in multiplication, satisfying \( E \cdot x = x \cdot E = x \) for any group element \( x \).
- The rules also include specific relationships like \( I^2 = -E \), \( J^2 = -E \), and \( K^2 = -E \).
- Rotation rules include combining elements, for example, \( I \cdot J = K \) which shows a property that \( Q \) elements are non-commutative.
