Question 4. Let G be a group. Define f : G → G via g → g¬1. (a) Prove that f is a bijection. (b) Prove that f is a homomorphism if and only if G is Abelain.
Question 4. Let G be a group. Define f : G → G via g → g¬1. (a) Prove that f is a bijection. (b) Prove that f is a homomorphism if and only if G is Abelain.
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
Related questions
Question
[Groups and Symmetries] How do you solve the fourth question 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F255aa298-2443-479d-9bde-3489e4fa5e8c%2Fb8c522bc-75f2-49b1-bcbd-91586a7ce869%2Fw05m7gm_processed.png&w=3840&q=75)
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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