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A (ii), (iii) (i), (ii) B P (1) (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}.

A (ii), (iii) (i), (ii) B P (1) (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}.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Suppose \( f: A \rightarrow B \) is a non-zero group homomorphism. Which of the following are true? (i) If \( A = \mathbb{Z}_{36}, B = \mathbb{Z}_{6}, f(2) = 4 \), then \( f(10) = 2 \). (ii) If \( A = B = \mathbb{Z}_{20}, f(7) = 9 \), then \( f(1) = 6 \). (iii) If \( f \) is an isomorphism, then \(\ker(f) \neq \{e\}\). (iv) If \( A = U_{10}, B = \mathbb{Z}_{4}, f(3) = 2 \), then \(\ker(f) = \{1\}\). Options: - A: (ii), (iii) - B: (i), (ii) - C: (i) - D: (iv) - E: (i), (iv)
Transcribed Image Text:Suppose \( f: A \rightarrow B \) is a non-zero group homomorphism. Which of the following are true? (i) If \( A = \mathbb{Z}_{36}, B = \mathbb{Z}_{6}, f(2) = 4 \), then \( f(10) = 2 \). (ii) If \( A = B = \mathbb{Z}_{20}, f(7) = 9 \), then \( f(1) = 6 \). (iii) If \( f \) is an isomorphism, then \(\ker(f) \neq \{e\}\). (iv) If \( A = U_{10}, B = \mathbb{Z}_{4}, f(3) = 2 \), then \(\ker(f) = \{1\}\). Options: - A: (ii), (iii) - B: (i), (ii) - C: (i) - D: (iv) - E: (i), (iv)
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