Question 3 with proof if it is a group.Abstract Algebra 1. Exercise 11 in §1.3. Let o is the m-cycle (1 2 ... m). Show that o' is also an m-cycle if and only ifi is relatively prime to m.2. Exercise 1 in §1.4. Show that if n is not prime then Z/nZ is not a field.3. Let n EN be given. Is the set U = {A : det A =±1} C Mat„xn(R) a group under matrix multipli-cation?4. Exercise 1 in §1.5. Compute the order of each of the elements in Qs-5. Let G, H be groups. Suppose that p: G →Hlisan isomorphism. Prove that : H →G is alsoan isomorphism.6. Exercise 1 in §1.6. Let G, H be groups. Let y: G H be a homomorphism.(a) Prove that p(a") = y(x)" for all n E N.(b) Do part (a) for n = -1 and deduce that p(r") = p(x)" for all n e Z.%3D- H is an isomomorphism, prove p(r)| = r|7. Exercise 2 in §1.6. Let G, H be groups. If y: Gfor r E G. Deduce that any two isomorphic groups hav ethe same number of elements of ordern foreach n e N. Is the result true if p is only assumed to be a homomorphism?%3D8. Exercise 17 in $1.6. Let G be any group. Prove that the map from G to itself defined by g g isa homomorphism if and only if G is abelian.LEGO

By using properties of group we solve the question no. 3 as follows :

Answered: 3. Let n eN be given. Is the set U =…

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3. Let n eN be given. Is the set U = {A: det A = ±1} C Matnxn(R) a group under matrix multipli- %3D cation?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 24E: If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type...

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Question

Question 3 with proof if it is a group. Abstract Algebra

1. Exercise 11 in §1.3. Let o is the m-cycle (1 2 ... m). Show that o' is also an m-cycle if and only if
i is relatively prime to m.
2. Exercise 1 in §1.4. Show that if n is not prime then Z/nZ is not a field.
3. Let n EN be given. Is the set U = {A : det A =
±1} C Mat„xn(R) a group under matrix multipli-
cation?
4. Exercise 1 in §1.5. Compute the order of each of the elements in Qs-
5. Let G, H be groups. Suppose that p: G →
Hlis
an isomorphism. Prove that : H →G is also
an isomorphism.
6. Exercise 1 in §1.6. Let G, H be groups. Let y: G H be a homomorphism.
(a) Prove that p(a") = y(x)" for all n E N.
(b) Do part (a) for n = -1 and deduce that p(r") = p(x)" for all n e Z.
%3D
- H is an isomomorphism, prove p(r)| = r|
7. Exercise 2 in §1.6. Let G, H be groups. If y: G
for r E G. Deduce that any two isomorphic groups hav ethe same number of elements of ordern for
each n e N. Is the result true if p is only assumed to be a homomorphism?
%3D
8. Exercise 17 in $1.6. Let G be any group. Prove that the map from G to itself defined by g g is
a homomorphism if and only if G is abelian.
LEGO

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