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 + x² 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 + x² is an isomorphism. n <0, and |g| # 2 for all g E G. Prove that the map ¢: G → G
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 32EQ
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