Question

Find the normalizer of the indicated subgroup in the indicated group. $$ \langle\mathrm{j}\rangle \text { in } Q_{8} $$

Short answer

The normalizer of \( \langle j \rangle \) in \( Q_8 \) is \( \{ 1, -1, j, -j \} \).

Step-by-step solution

Understand the Quaternion Group

The quaternion group, denoted as \( Q_8 \), is the group \( \{ 1, -1, i, -i, j, -j, k, -k \} \) with the multiplication rules \( i^2 = j^2 = k^2 = ijk = -1 \). It is a non-abelian group of order 8.

Define Subgroup \( \langle j \rangle \)

The subgroup \( \langle j \rangle \) is generated by the element \( j \) from \( Q_8 \). Since \( j^2 = -1 \), the subgroup \( \langle j \rangle \) contains the elements \( \{ 1, j, -1, -j \} \).

Define the Normalizer

The normalizer of a subgroup \( H \) in a group \( G \), denoted by \( N_G(H) \), is the set of elements in \( G \) that conjugate \( H \) into itself. Formally, \( N_G(H) = \{ g \in G \mid gHg^{-1} = H \} \).

Find Conjugates in \( Q_8 \)

Conjugating element \( j \) by another element \( g \in Q_8 \) requires finding \( gjg^{-1} \). Compute for each element: - For \( 1 \) and \( -1 \): \( 1j1^{-1} = j \), \( (-1)j(-1)^{-1} = j \).- For \( i, -i \): \( iji^{-1} = -j \), \( (-i)j(-i)^{-1} = -j \).- For \( j, -j \): \( jjj^{-1} = j \), \( (-j)j(-j)^{-1} = j \).- For \( k, -k \): \( kjk^{-1} = -j \), \( (-k)j(-k)^{-1} = -j \).

Determine the Normalizer

Based on calculations, those elements from \( Q_8 \) for which \( gjg^{-1} \) results in elements of \( \langle j \rangle \) (i.e., \( j \) or \( -j \)) are \( N_{Q_8}(\langle j \rangle) = \{ 1, -1, j, -j \} \).

Validate Normalizer Contains Subgroup

We confirm that the normalizer \( N_{Q_8}(\langle j \rangle) \) properly contains the subgroup \( \langle j \rangle = \{ 1, j, -1, -j \} \), as all elements in \( \langle j \rangle \) conjugate \( j \) to other elements in \( \langle j \rangle \).

Key concepts

Quaternion Group

The Quaternion group, often denoted by \( Q_8 \), is an essential concept in group theory, specifically in the study of non-abelian groups. It comprises eight elements: \( \{ 1, -1, i, -i, j, -j, k, -k \} \). These elements follow specific multiplication rules:

• \( i^2 = j^2 = k^2 = -1 \)
• \( ijk = -1 \)

One significant aspect of \( Q_8 \) is its non-abelian nature. This means that the order in which you multiply elements matters; for instance, \( ij eq ji \). This characteristic makes it a fascinating subject of study for those learning about advanced algebraic structures. The quaternion group is small, of order 8, providing a simple yet profound example of complex group behavior. It is also used in various mathematical applications, including in the fields of 3D rotations and spatial transformations.

Group Theory

Group theory is a branch of mathematics that studies groups, which are sets equipped with an operation that combines any two elements to form a third element within the set, subject to four main properties: closure, associativity, identity, and invertibility. Here's a quick rundown:

• Closure: For any elements \( a \) and \( b \) in the group, the result of the operation, \( a \ast b \), is also in the group.
• Associativity: For any elements \( a, b, \) and \( c \) in the group, \((a \ast b) \ast c = a \ast (b \ast c)\).
• Identity Element: There is an element \( e \) in the group such that for any element \( a \) in the group, \( e \ast a = a \ast e = a \).
• Inverse: For each element \( a \) in the group, there is an element \( b \) in the group such that \( a \ast b = b \ast a = e \).

Group theory is foundational for many areas of mathematics and the sciences. It allows for the classification and study of symmetry, provides insights into the algebraic structure of objects, and serves as a bridge to more advanced theories like ring and field theory, which extend these concepts further.

Normalizer

The concept of a normalizer within group theory is particularly intriguing, and it refers to the set of elements in a group that keeps a given subgroup invariant under conjugation. Formally, for a subgroup \( H \) of a group \( G \), the normalizer, denoted \( N_G(H) \), includes elements \( g \) in \( G \) such that: \[ gHg^{-1} = H \]

• This means that when you conjugate each element of the subgroup by some fixed element \( g \), you end up with the subgroup itself.
• The normalizer is always a subgroup of \( G \) that includes \( H \) as a subset.

The normalizer is important because it provides insight into the structure of \( G \) regarding \( H \) and helps in understanding the overall symmetry of the group. It acts as a sort of maximal stabilizer, showing how elements of \( G \) can transform \( H \) while keeping its structure intact. This concept is also crucial in analyzing group actions and understanding the relationships between different subgroups.

Subgroup

A subgroup is a subset of a group that is itself a group under the operation defined on the larger set. For instance, in the quaternion group \( Q_8 \), the set \( \langle j \rangle = \{ 1, j, -1, -j \} \) is a subgroup generated by the element \( j \).

• Any subgroup must satisfy the group properties: closure, associativity, identity, and inverses.
• Subgroups often reveal important structural aspects of the larger group they are part of.

Subgroups play a critical role in understanding the internal structure of more complex groups, as they can help identify invariant features and serve as building blocks for the entire group. In solving problems related to groups, recognizing subgroups can simplify understanding and operations within the parent group, thus providing clarity into the group's characteristics.