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Let G be the group of all transformations on R which have the form x → ax + b where a, b e R, a ± 0. (i) Prove H = {x- > ах + b | а — 1} is a normal subgroup of G. Describe the right and left cosets of H to G. (iii) Is the subset K = {x → ax} a normal subgroup of G? (iv) Is G abelian? Is H abelian? (ii)

Let G be the group of all transformations on R which have the form x → ax + b where a, b e R, a ± 0. (i) Prove H = {x- > ах + b | а — 1} is a normal subgroup of G. Describe the right and left cosets of H to G. (iii) Is the subset K = {x → ax} a normal subgroup of G? (iv) Is G abelian? Is H abelian? (ii)

Advanced Engineering Mathematics
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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please just answer iv. thank u

**Group Transformations on \(\mathbb{R}\)** Let \( G \) be the group of all transformations on \(\mathbb{R}\) which have the form \( x \to ax + b \) where \( a, b \in \mathbb{R}, a \neq 0 \). 1. **Prove \( H = \{ x \to ax + b \mid a = 1 \} \) is a normal subgroup of \( G \).** 2. **Describe the right and left cosets of \( H \) to \( G \).** 3. **Is the subset \( K = \{ x \to ax \} \) a normal subgroup of \( G \)?** 4. **Is \( G \) abelian? Is \( H \) abelian?**
Transcribed Image Text:**Group Transformations on \(\mathbb{R}\)** Let \( G \) be the group of all transformations on \(\mathbb{R}\) which have the form \( x \to ax + b \) where \( a, b \in \mathbb{R}, a \neq 0 \). 1. **Prove \( H = \{ x \to ax + b \mid a = 1 \} \) is a normal subgroup of \( G \).** 2. **Describe the right and left cosets of \( H \) to \( G \).** 3. **Is the subset \( K = \{ x \to ax \} \) a normal subgroup of \( G \)?** 4. **Is \( G \) abelian? Is \( H \) abelian?**
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