Abstract Algebra 5. If gcd(m, n) = d, then there are d homomorphisms from Zm to Zn. Identifythe four homomorphisms from Z12 to Z20Hint: If y(1) = y, why must the congruence my = 0 mod n be satisfied?

Given: If gcd(m, n)=d then there are d homomorphisms from ℤm to ℤn. To find: Four homomorphisms from…

Answered: 5. If gcd(m, n) = d, then there are d…

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

Let be as described in the proof of Theorem. Give a specific example of a positive element of .
Write 20 as the direct sum of two of its nontrivial subgroups.
11. Show that defined by is not a homomorphism.
Find all monic irreducible polynomials of degree 2 over Z3.
6. Prove that if is a permutation on , then is a permutation on .
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 ]
1. Consider , the groups of units in under multiplication. For each of the following subgroups in , partition into left cosets of , and state the index of in a. b.
2. List all the elements of (Z30, Đ) that are of order 15.
Negative points per question: Based on Section 4.2 of HMU. 1. h is a homomorphism from the alphabet {a,b,c} to {0,1}. If h(a) = 01, h(b) = 0, and h(c) = 10, which of the following strings is in h¯¹ (010010)? a) babc b) bacb c) cbbc d) abbc 2. The language of regular expression (0+10)* is the set of all strings of O's and 1's such that every 1 is immediately followed by a 0. Describe the complement of this language (with respect to the alphabet {0,1}) and identify in the list below the regular expression whose language is the complement of L((0+10)*). a) (0+1)*11(0+1)* + (0+10)*1 b) (0+10)*11(0+1)* c) (1+01)* d) (0+10)*1(ɛ+11(0+1)*) 3. If h is the homomorphism defined by h(a) = 0 and h(b) = ε, which of the following strings is in h¨¹ (000)? a) abaabaa b) abbbabaab c) abaabbb d) babab 4. Find, in the list below, a regular expression whose language is the reversal of the language of this regular expression: 10(2+3). Recall that the reversal of a language is formed by reversing all its…
4.6 List all the elements of (Z45, ) that are of order 15.
1. (a) Write down the formulas for all homomorphisms from Z10 into Z25. (b) Write down the formulas for all homomorphisms from Z24 into Z18.
1. List a pair of zero divisors of (M₂(Z), +, ×): ol ]

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

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

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

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

5. If gcd(m, n) = d, then there are d homomorphisms from Zm to Zn. Identify the four homomorphisms from Z12 to Z20