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Question 5. Suppose G is an Abelain group, |G| = n < ∞, and |g| # 2 for all g E G. Prove that the map : G → G given by x + x² is an isomorphism.

Question 5. Suppose G is an Abelain group, |G| = n < ∞, and |g| # 2 for all g E G. Prove that the map : G → G given by x + x² is an isomorphism.

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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[Groups and Symmetries] How do you solve question 5? 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 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 p(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 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 → G given by x + x² is an isomorphism.
Transcribed Image Text: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 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 p(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 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 → G given by x + x² is an isomorphism.
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