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

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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[Groups and Symmetries] How would you solve  2 thanks a lot

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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