6. Describe the left cosets of SL2(R) in GL₂(R). What is the index of SL₂(R) in GL₂(R)?
6. Describe the left cosets of SL2(R) in GL₂(R). What is the index of SL₂(R) in GL₂(R)?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**6. Describe the left cosets of \( SL_2(\mathbb{R}) \) in \( GL_2(\mathbb{R}) \). What is the index of \( SL_2(\mathbb{R}) \) in \( GL_2(\mathbb{R}) \)?**
In this problem, we are asked to explore the relationship between two important groups in linear algebra: the special linear group \( SL_2(\mathbb{R}) \) and the general linear group \( GL_2(\mathbb{R}) \).
Firstly, the special linear group \( SL_2(\mathbb{R}) \) consists of all \( 2 \times 2 \) real matrices with determinant equal to 1. The general linear group \( GL_2(\mathbb{R}) \), on the other hand, consists of all invertible \( 2 \times 2 \) real matrices, i.e., those with non-zero determinants.
The concept of a left coset is central to this question. For a group \( G \) and a subgroup \( H \subset G \), the left coset of \( H \) in \( G \) with respect to an element \( g \in G \) is given by:
\[ gH = \{ gh \mid h \in H \}. \]
We are asked to describe such left cosets of \( SL_2(\mathbb{R}) \) in \( GL_2(\mathbb{R}) \).
Furthermore, the index of a subgroup \( H \) in \( G \), denoted \( [G : H] \), is the number of distinct left cosets of \( H \) in \( G \). In this context, it represents the "size" of \( GL_2(\mathbb{R}) \) in terms of \( SL_2(\mathbb{R}) \).
To calculate this index, consider that every matrix in \( GL_2(\mathbb{R}) \) can be expressed as a product of a scalar matrix (a diagonal matrix with the same scalar on the diagonal) and a matrix from \( SL_2(\mathbb{R}) \). Therefore, the cosets can be parameterized by the determinant value (up to an absolute value scaling), indicating that the index is infinite. This reflects the continuous nature and infinite size difference between these two groups.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6389447-1237-4af0-b5c6-eb1260425b55%2F50a8d62b-939f-4c2c-80b9-9696b7b8ce7f%2Frzxmxf_processed.png&w=3840&q=75)
Transcribed Image Text:**6. Describe the left cosets of \( SL_2(\mathbb{R}) \) in \( GL_2(\mathbb{R}) \). What is the index of \( SL_2(\mathbb{R}) \) in \( GL_2(\mathbb{R}) \)?**
In this problem, we are asked to explore the relationship between two important groups in linear algebra: the special linear group \( SL_2(\mathbb{R}) \) and the general linear group \( GL_2(\mathbb{R}) \).
Firstly, the special linear group \( SL_2(\mathbb{R}) \) consists of all \( 2 \times 2 \) real matrices with determinant equal to 1. The general linear group \( GL_2(\mathbb{R}) \), on the other hand, consists of all invertible \( 2 \times 2 \) real matrices, i.e., those with non-zero determinants.
The concept of a left coset is central to this question. For a group \( G \) and a subgroup \( H \subset G \), the left coset of \( H \) in \( G \) with respect to an element \( g \in G \) is given by:
\[ gH = \{ gh \mid h \in H \}. \]
We are asked to describe such left cosets of \( SL_2(\mathbb{R}) \) in \( GL_2(\mathbb{R}) \).
Furthermore, the index of a subgroup \( H \) in \( G \), denoted \( [G : H] \), is the number of distinct left cosets of \( H \) in \( G \). In this context, it represents the "size" of \( GL_2(\mathbb{R}) \) in terms of \( SL_2(\mathbb{R}) \).
To calculate this index, consider that every matrix in \( GL_2(\mathbb{R}) \) can be expressed as a product of a scalar matrix (a diagonal matrix with the same scalar on the diagonal) and a matrix from \( SL_2(\mathbb{R}) \). Therefore, the cosets can be parameterized by the determinant value (up to an absolute value scaling), indicating that the index is infinite. This reflects the continuous nature and infinite size difference between these two groups.
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