Question

Determine whether the indicated subgroup is normal in the indicated group. For \(r \in \mathbb{R}^{*}\) let \(r I=\left[\begin{array}{rr}r & 0 \\ 0 & r\end{array}\right]\). Show that \(H=\left\\{r l \mid r \in \mathbb{R}^{*}\right\\}\) is a normal subgroup of \(\mathrm{GL}(2, \mathbb{R})\)

Short answer

Yes, \(H\) is a normal subgroup of \(\mathrm{GL}(2, \mathbb{R})\).

Step-by-step solution

Define Normal Subgroup

A subgroup \(H\) of a group \(G\) is normal if for every element \(g\) in \(G\), the conjugate \(gHg^{-1}\) is a subset of \(H\). In symbols, \(H \trianglelefteq G\) if \(gHg^{-1} \subseteq H, \forall g \in G\).

Define Groups and Elements

The group \(\mathrm{GL}(2, \mathbb{R})\) consists of all invertible \(2 \times 2\) matrices with real entries. The subgroup \(H\) consists of matrices of the form \(r I = \begin{bmatrix} r & 0 \ 0 & r \end{bmatrix}\), where \(r \in \mathbb{R}^{*}\) (the set of non-zero real numbers).

Conjugate an Element of H

Consider an arbitrary element \(A \in \mathrm{GL}(2, \mathbb{R})\) and an arbitrary element \(h = rI \in H\). We need to check that \(A(h)A^{-1} = A(rI)A^{-1} = (ArI)A^{-1}\).

Calculate Conjugate and Show It Belongs to H

The expression for the conjugation is:\[A(rI)A^{-1} = A\begin{bmatrix} r & 0 \ 0 & r \end{bmatrix}A^{-1} = r(AA^{-1}) = rI\]This shows that \(A(h)A^{-1} = rI\), which is an element of \(H\). All conjugated matrices still lie in \(H\).

Conclusion of Normality

Since \(A\) was an arbitrary element of \(\mathrm{GL}(2, \mathbb{R})\), the conjugation \(A h A^{-1} \in H\) holds for all \(A\). Therefore, \(H\) is normal in \(\mathrm{GL}(2, \mathbb{R})\).

Key concepts

General Linear Group

The General Linear Group, denoted as \( \mathrm{GL}(n, \mathbb{R}) \), is a fundamental algebraic structure in linear algebra. It consists of all \( n \times n \) invertible matrices with real number entries. Understanding this group is crucial because it represents linear transformations that can map vectors to other vectors in \( n \)-dimensional space without collapsing them. This property of invertibility means each transformation has an inverse, making them particularly useful in many areas of mathematics and physics. Here are some key features of the General Linear Group:

• The determinant of an invertible matrix is non-zero, highlighting how these matrices affect volume scaling but not collapsing.
• Matrix multiplication serves as the group operation, where the identity element is the identity matrix \( I \), which maintains the properties of any vector it multiplies.
• This group is highly non-commutative, meaning matrix multiplication within it does not simply follow the commutative law \( AB eq BA \) in general.

The General Linear Group is a building block for exploring more complex structures, including those seen in geometry and change of basis calculations in vector spaces.

Matrix Groups

Matrix groups are collections of matrices that form a group under matrix multiplication. These groups must satisfy four fundamental group properties: closure, associativity, identity, and invertibility. Let's delve into how these properties apply:

• Closure: Multiplying any two matrices from the group results in another matrix that is also in the group.
• Associativity: For any three matrices \( A, B, \) and \( C \) in the group, the equation \( (AB)C = A(BC) \) holds.
• Identity: There exists an identity matrix \( I \) that leaves any matrix from the group unchanged when multiplied.
• Invertibility: Every matrix in the group has an inverse matrix within the group, ensuring the equation \( AA^{-1} = I \).

In the context of the problem, the subgroup \( H \) is a matrix group within \( \mathrm{GL}(2, \mathbb{R}) \). It consists of matrices that scale the identity matrix by a non-zero real number. This ensures each matrix, when multiplied by its inverse, returns the identity matrix \( I \), supporting the properties required of a matrix group.

Conjugation in Groups

Conjugation is an essential concept in group theory that deals with transforming one element of a group into another while still remaining within the same group through a specific process. This transformation is defined as \( gHg^{-1} \), where \( g \) is an element of the group \( G \) and \( H \) is a subgroup of \( G \). Here's a simpler breakdown:

• The Process: Take an element \( h \) from \( H \), and use an arbitrary element \( g \) from \( G \) to compute the product \( ghg^{-1} \).
• Result within Group: If the result of this conjugation is still within \( H \), then \( H \) is considered a normal subgroup of \( G \).
• Importance: This ability to convert elements internally while remaining within the subgroup underscores normality and helps identify how subgroups interact with the larger group structure.

Through conjugation, we're able to verify if a subgroup maintains its structural properties throughout the entire flip and transformation process. In matrix groups, this involves taking any matrix from the general group, applying similar transformations, and confirming the outcome still fits within the subgroup, as demonstrated in the solution.