DEADEAll(i), (iii), (iv)(ii), (iii), (iv)(i), (ii), (iv)NoneWhich of the following are group homomoprhisms?(i): : Z₂ → Z2, x + x² (mod 2).(ii): o: Z→ Z₂, x→x² (mod 2).(iii): o: Z→ Z3, 2 + x² (mod 3).(iv): o: Z→ Z3,xx³ (mod 3).:

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Answered: Which of the following are group…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.8: Some Results On Finite Abelian Groups (optional)

Problem 6E: 6. For each of the following values of , describe all the abelian groups of order , up to...

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A B D E (i), (ii) (ii), (iv) (iii), (iv) All None Which of the following are ring homomorphisms? (i): f: (2Z, +,-) → (Z,+,-), 3x. (ii): f: (Z8, +,-) → (Z₁, +,-), 2x – 1 (mod 4). (iii): f: (Z8, +,-) → (Z6, +,-), x4x (mod 6). (iv): f: (Z2 x Z5, +,-) → (Z5, +,-), (x,y) →x+y (mod 5) :
B (ii), (iii) (i), (ii) (i) (iv) (i), (iv) Suppose f: A B is a non-zero group homomorphism. Which of the following are true? (i): If A=Z36, B = Ze, f(2)=4, then f(10) = 2. (ii): If A = B = Z20, f (7) = 9, then f(1) = 6. (iii): If f is an isomorphism, then ker(f) # {e}. (iv): If A = U10, B = Z4, f(3) = 2, then ker(f) = {1}.
A B D E All (ii), (iii) None (i), (iii) (i), (ii) Suppose such that (i): (ii): : Z→ Z40 is a group homomorphism and ker() = 20Z (12) = 8. Which of the following are true? (32) = 23. (-32) = 32. (iii): (52) = 8. (iv): p(-52) = 23. ...
8 1 C A B (ii), (iii) (i), (ii) (i) (iv) (i), (iv) Suppose f: A B is a non-zero group homomorphism. Which of the following are true? (i): If A = Z36, B = Z6, f(2)=4, then f(10) = 2. (ii): If A = B = Z20, f (7) = 9, then f(1) = 6. (iii): If f is an isomorphism, then ker(f)# {e}. (iv): If A = U10, B = Z₁, f(3) = 2, then ker(f) = {1}.
B E None (i), (ii) (iii), (iv) Which of the following are ring homomorphisms? (i): f: (Q(√2), +,-) → (Q(√3), +,-), a +b√2+a+b√3. (ii): f: C Rx R, a +bi (a, b). (iii): f: (QxQ, +,-) → (Q(√2), +,-), (x,y) →x+yv2. (iv): f: CR, a +bi+ a² + b². ...
3. Let f : (R\{0},-) → (R\{0}, ·) be group homomorphism defined by f(a) = |al. Then ker(f) = %3D
A B (iv) © (i), (ii) (i), (iv) (ii), (iii) Suppose f: A→ B is a non-zero group homomorphism. Which of the following are true? (i): If A = Z36, B = Z6, f(2)= 4, then f(10) = 2. (ii): If A = B = Z20, f (7)= 9, then f(1) = 6. (iii): If f is an isomorphism, then ker(f)# {e}. (iv): If A = U10, B = Z₁, f(3) = 2, then ker(f) = {1}.
B (i), (iii), (iv) (ii), (iii), (iv) (i), (ii), (iv) None Which of the following are group homomoprhisms? (i): o: Z₂ → Z₂, 2x² (mod 2). (ii): o: Z→ Z₂, ² (mod 2). (iii): o: Z→ Z3, xx² (mod 3). (iv): : Z→ Z3,2³ (mod 3).

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Which of the following are group homomoprhisms? (i): : Z₂ → Z₂, x + x² (mod 2). (ii): o: Z→ Z₂, x² (mod 2). (iii): o: Z→ Z3, x + x² (mod 3). (iv): o: Z→ Z3,2 x³ (mod 3).