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Magazine Chapter 3: Problem 15
$$ \text { Find a subgroup of } \mathbb{Z}_{4} \times Q_{8} \text { that is not normal, where } Q_{8} \text { is the quaternion group. } $$
Short Answer
Expert verified
The subgroup \(H = \{(0, 1), (0, -1)\}\) is not normal in \(\mathbb{Z}_4 \times Q_8\).
Step by step solution
01
Understanding the Groups
First, let's identify the groups involved. The group \( \mathbb{Z}_{4} \) is the cyclic group with four elements: {0, 1, 2, 3} under addition modulo 4. The quaternion group \( Q_{8} \) consists of the elements {1, -1, i, -i, j, -j, k, -k} under quaternion multiplication.
02
Identifying the Direct Product
The group \( \mathbb{Z}_{4} \times Q_{8} \) is the set of ordered pairs \((a, b)\) where \(a \in \mathbb{Z}_{4}\) and \(b \in Q_{8}\). It consists of \(4 \times 8 = 32\) elements with operation defined as \((a, b) \cdot (c, d) = (a + c, b \cdot d)\).
03
Searching for a Subgroup
A subgroup is a subset that is closed under the group operation and contains the identity element of the original group. Consider the subgroup \(H = \{(0, 1), (0, -1)\}\), which is formed by selecting a subgroup from the second component \(Q_8\) that is not the entire group.
04
Check Subgroup Closure
First, verify that \(H = \{(0, 1), (0, -1)\}\) is a subgroup: the identity element \((0, 1)\) is in \(H\). Also, this subgroup is closed under the operation because \((0, 1) \cdot (0, -1) = (0, -1)\) and \((0, -1) \cdot (0, -1) = (0, 1)\).
05
Verify H is Not Normal
To determine if \(H\) is normal, check if for every \((a, b) \in \mathbb{Z}_4 \times Q_8\), the set \((a, b)H = H(a, b)\) holds true. Find an element such as \((1, i)\) where the conjugate \((1, i)(0, -1)(-1, -i)\) is outside \(H\), confirming \(H\) is not normal.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quaternion Group
The quaternion group, denoted as \( Q_8 \), is a fascinating example of a non-abelian group, which means the group operation does not necessarily commute: \( ab eq ba \). This group consists of eight elements: \{1, -1, i, -i, j, -j, k, -k\}. It is related to certain rotations, specifically in three-dimensional space.
- Elements of the Group: Its elements include 1 (the identity element), and three pairs of units: i, j, and k each paired with their negatives.
- Multiplication Rules: The multiplication rules are based on quaternion multiplication, where \(i^{2} = j^{2} = k^{2} = -1\), and \( ij = k, ji = -k \) (note the non-commutativity), \( jk = i, kj = -i\), \( ki = j, ik = -j\).
- Identity Element: The group has 1 as the identity element, as it does not change any quaternion when multiplied.
Cyclic Group
A cyclic group is one of the simplest types of groups in abstract algebra. It is generated by a single element, meaning every other element in the group can be expressed as a power of this generator.
- Examples: \( \mathbb{Z}_4 \) is a cyclic group consisting of elements \{0, 1, 2, 3\}. The operation is addition modulo 4. Every element in this group can be obtained by repeated addition of 1, which is its generator.
- Properties: All cyclic groups are abelian, meaning their group operation commutes: \( ab = ba \). They are incredibly useful for their simplicity and are foundational to the structure of more complex groups.
- Significance: Cyclic groups serve as building blocks in group theory. They help us understand more complicated and higher-order groups through their simple structure.
Direct Product
In group theory, the concept of the direct product allows the construction of a new group from two given groups. This operation combines two sets of elements into pairs, with each pair's operation being defined by the original groups' operations on their respective components.
- Constructing a Direct Product: To create the direct product \( \mathbb{Z}_4 \times Q_8 \), consider all possible ordered pairs \((a, b)\) where \(a\) is from \( \mathbb{Z}_4 \) and \(b\) is from \( Q_8 \). Thus, there are 4 elements from \( \mathbb{Z}_4 \) and 8 from \( Q_8 \), resulting in a total of 32 elements.
- Operation Rule: The operation on these pairs is defined by \((a, b) \cdot (c, d) = (a + c, b \cdot d)\). This operation respects the operations of \( \mathbb{Z}_4 \) and \( Q_8 \).
- Applications: Direct products are important for decomposing groups and analyzing their structure, allowing complex systems to be viewed as combinations of simpler systems.
Subgroup
In abstract algebra, a subgroup is a subset of a group that forms a group itself under the same operation. Identifying subgroups helps in understanding the larger group's structure.
- Criteria for Subgroup: A subgroup must satisfy three criteria:
- Contain the identity element of the larger group.
- Be closed under the group's operation, meaning any operation on elements within the subgroup results in an element still within the subgroup.
- Every element must have an inverse in the subgroup.
- Example: In the context \( \mathbb{Z}_4 \times Q_8 \), consider the subgroup \( H = \{(0, 1), (0, -1)\} \). It shows closure and contains the identity, but is not normal, which is determined by checking conjugacy within the larger group.
- Normal Subgroup: A normal subgroup requires each element \( g \) of the larger group to satisfy \( gH = Hg \). If this is not true for all elements, the subgroup is not normal.
