[Groups and Symmetries] How do you solve question 3? Thanks Question 1. Show that in S7, the equation r²(1234) has no solutions.Question 2. Let n be an even positive integer. Prove that An has an element of order greater than n if and only ifn > 8.Question 3. Notice that the set {1,–1} is a group under multiplication. Fix n > 2. Define p : Sn →{1, –1} viaif o is an even permutationp(0) =1if o is an odd permutationProve that y is a group homomorphism. Also compute ker y.Question 4. Let G be a group. Define f :G → G via g Hg¬1.(a) Prove that f is a bijection.(b) Prove that f is a homomorphism if and only if G is Abelain.Question 5. Suppose G is an Abelain group, |G| = n < ∞, and |g| #2 for all g e G. Prove that the map ø : G → Ggiven by x + x² is an isomorphism.

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Question 1. Show that in S7, the equation x2 (1234) has no solutions. Question 2. Let n be an even positive integer. Prove that An has an element of order greater than n if and only if n > 8. Question 3. Notice that the set {1, –1} is a group under multiplication. Fix n > 2. Define p : Sn → {1, –1} via - if o is an even permutation e(0) = -1 if o is an odd permutation Prove that y is a group homomorphism. Also compute ker y. Question 4. Let G be a group. Define f : G → G via g → g1. (a) Prove that f is a bijection. (b) Prove that ƒ is a homomorphism if and only if G is Abelain. Question 5. Suppose G is an Abelain group, |G| = given by x + æ² is an isomorphism. n <0, and |g| # 2 for all g E G. Prove that the map ¢: G → G

Question 1. Show that in S7, the equation x2 (1234) has no solutions. Question 2. Let n be an even positive integer. Prove that An has an element of order greater than n if and only if n > 8. Question 3. Notice that the set {1, –1} is a group under multiplication. Fix n > 2. Define p : Sn → {1, –1} via - if o is an even permutation e(0) = -1 if o is an odd permutation Prove that y is a group homomorphism. Also compute ker y. Question 4. Let G be a group. Define f : G → G via g → g1. (a) Prove that f is a bijection. (b) Prove that ƒ is a homomorphism if and only if G is Abelain. Question 5. Suppose G is an Abelain group, |G| = given by x + æ² is an isomorphism. n <0, and |g| # 2 for all g E G. Prove that the map ¢: G → G

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 3E: In Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its...

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