1. Let H = {β ∈ S5 : β(4) = 4}. Prove that H is a subgroup of S5. (Reminder: The group operation of S5 is composition of functions, and it’s helpful here to use the definition (αβ)(n) = α(β(n)).)

1. Let H = {β ∈ S5 : β(4) = 4}. Prove that H is a subgroup of S5. (Reminder: The group operation of S5 is composition of functions, and it’s helpful here to use the definition (αβ)(n) = α(β(n)).)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 2E: For each of the following subgroups H of the addition groups Z18, find the distinct left cosets of H...

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Question

1. Let H = {β ∈ S5 : β(4) = 4}. Prove that H is a subgroup of S5.
(Reminder: The group operation of S5 is composition of functions, and it’s helpful here to use the definition (αβ)(n) = α(β(n)).)

2. Show that Z6 is isomorphic to U(7) by giving an isomorphism φ from Z6 to U(7).

3.Let H = { [ 1 a ]: a ∈ Z} .Define a map φ : Z →H by

[ 0 1 ]

φ(n)= [ 1 n ]

[ 0 1 ]

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