Let : Z₁₂ → Z, be the homomorphism such that (1) = 2.12(a) State the First Isomorphism Theorem.(b) Find the kernel K of .(e) List the cosets in Z₁/K. showing the elements in each coset.(d) Give the correspondence between 2₁./K and Z, given by the map described in thetheorem in (a).

Answered: Image /qna-images/answer/c239bb77-9b54-4c08-b56f-446204d33537.jpg

Answered: Let :Z₁₂ →Z, be the homomorphism such…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 32E: 32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping ...

See similar textbooks

Similar questions

Let G=1,i,1,i under multiplication, and let G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. Find an isomorphism from G to G that is different from the one given in Example 5 of this section. Example 5 Consider G=1,i,1,i under multiplication and G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. In order to define a mapping :G4 that is an isomorphism, one requirement is that must map the identity element 1 of G to the identity element [ 0 ] of 4 (part a of Theorem 3.30). Thus (1)=[ 0 ]. Another requirement is that inverses must map onto inverses (part b of Theorem 3.30). That is, if we take (i)=[ 1 ] then (i1)=((i))1=[ 1 ] Or (i)=[ 3 ] The remaining elements 1 in G and [ 2 ] in 4 are their own inverses, so we take (1)=[ 2 ]. Thus the mapping :G4 defined by (1)=[ 0 ], (i)=[ 1 ], (1)=[ 2 ], (i)=[ 3 ]
9. For any let denote in and let denote in . a. Prove that the mapping defined by is a homomorphism. b. Find ker .
28. For each, define by for. a. Show that is an element of . b. Let .Prove that is a subgroup of under mapping composition. c. Prove that is abelian, even though is not.
Let G be a group. Prove that the relation R on G, defined by xRy if and only if there exist an aG such that y=a1xa, is an equivalence relation. Let xG. Find [ x ], the equivalence class containing x, if G is abelian. (Sec 3.3,23) Sec. 3.3, #23: 23. Let R be the equivalence relation on G defined by xRy if and only if there exists an element a in G such that y=a1xa. If x(G), find [ x ], the equivalence class containing x.
Exercises 22. Let be a finite cyclic group of order with generators and . Prove that the mapping is an automorphism of .
Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
Write 20 as the direct sum of two of its nontrivial subgroups.
14. Let be an abelian group of order where and are relatively prime. If and , prove that .
11. Show that defined by is not a homomorphism.
For fixed integers a and b, let S={ ax+byxandy }. Prove that S is a subgroup of under addition.(A special form of this S is used in proving the existence of a greatest common divisor in Theorem 2.12.)
Exercises 18. Suppose and let be defined by . Prove or disprove that is an automorphism of the additive group .
Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

Let :Z₁₂ →Z, be the homomorphism such that (1) = 2. (a) State the First Isomorphism Theorem. (b) Find the kernel K of . (c) List the cosets in Z₁/K, showing the elements in each coset. (d) Give the correspondence between Z₁/K and Z, given by the map u described in the theorem in (a).