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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 → ax + b | a = 1} is a normal subgroup of G. (ii) 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?

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 → ax + b | a = 1} is a normal subgroup of G. (ii) 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?

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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I know that you can only answer 3, so you can at least solve just the first 3. Thank you. 

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