Let $ G = \{ f : [0,1] \rightarrow [0,1] \mid f \text{ is a bijection} \} $ be the group of bijections on $[0,1]$ with composition as the operation. For any $ a \in G$, let $ H_a = \{ f \in G \mid f(a) = a \} $. Show that if $ g \in G$, then $ gH_ag^{-1} = H_{g(a)} $. Is $ H_a $ a normal subgroup of $ G $?

Here we use the property that if forms a group under composition.

Let G = {f : [0, 1] → [0, 1] | ƒ is a bijection} be the group of bijections on [0, 1] with composition as the operation. For any a e G, let Ha = {f € G| f(a) = a}. Show that if g € G, then 9H@9¬ = Hg(a). Is Ha a normal subgroup of G? %3D

Let G = {f : [0, 1] → [0, 1] | ƒ is a bijection} be the group of bijections on [0, 1] with composition as the operation. For any a e G, let Ha = {f € G| f(a) = a}. Show that if g € G, then 9H@9¬ = Hg(a). Is Ha a normal subgroup of G? %3D

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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Let $ G = \{ f : [0,1] \rightarrow [0,1] \mid f \text{ is a bijection} \} $ be the group of bijections on $[0,1]$ with composition as the operation. For any $ a \in G$, let $ H_a = \{ f \in G \mid f(a) = a \} $. Show that if $ g \in G$, then $ gH_ag^{-1} = H_{g(a)} $. Is $ H_a $ a normal subgroup of $ G $?

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