1. Prove directly that there cannot exist a homomorphism : Q8 → S5 suchthat (j) = (1234) and (k) = (5314).

Answered: Image /qna-images/answer/0221f841-5e3a-4a7e-a3d1-9c0d58dedca2.jpg

Answered: Prove directly that there cannot exist…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.2: Ring Homomorphisms

Problem 11E: 11. Show that defined by is not a homomorphism.

See similar textbooks

Similar questions

11. Show that defined by is not a homomorphism.
Prove that if f is a permutation on A, then (f1)1=f.
31. Prove statement of Theorem : for all integers and .
Let be as described in the proof of Theorem. Give a specific example of a positive element of .
Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.
Find all monic irreducible polynomials of degree 2 over Z3.
Write 20 as the direct sum of two of its nontrivial subgroups.
12. Consider the mapping defined by . Decide whether is a homomorphism, and justify your decision.
9. For any let denote in and let denote in . a. Prove that the mapping defined by is a homomorphism. b. Find ker .
23. Let be the equivalence relation on defined by if and only if there exists an element in such that .If , find , the equivalence class containing.
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 ]
3. For each of the following mappings, write out and for the given and, where.

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

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

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

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS

Prove directly that there cannot exist a homomorphism : Q8 that (j) = (1234) and (k) = (5314). S5 such