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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, xx² (mod 3). (iv): o: Z→ Z3,2x³ (mod 3).

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, xx² (mod 3). (iv): o: Z→ Z3,2x³ (mod 3).

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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**Question:** Which of the following are group homomorphisms? (i) \( \phi : \mathbb{Z}_2 \to \mathbb{Z}_2, \, x \mapsto x^2 \, (\text{mod} \, 2) \). (ii) \( \phi : \mathbb{Z} \to \mathbb{Z}_2, \, x \mapsto x^2 \, (\text{mod} \, 2) \). (iii) \( \phi : \mathbb{Z} \to \mathbb{Z}_3, \, x \mapsto x^2 \, (\text{mod} \, 3) \). (iv) \( \phi : \mathbb{Z} \to \mathbb{Z}_3, \, x \mapsto x^3 \, (\text{mod} \, 3) \). **Options:** A. All B. (i), (iii), (iv) C. (ii), (iii), (iv) D. (i), (ii), (iv) E. None
Transcribed Image Text:**Question:** Which of the following are group homomorphisms? (i) \( \phi : \mathbb{Z}_2 \to \mathbb{Z}_2, \, x \mapsto x^2 \, (\text{mod} \, 2) \). (ii) \( \phi : \mathbb{Z} \to \mathbb{Z}_2, \, x \mapsto x^2 \, (\text{mod} \, 2) \). (iii) \( \phi : \mathbb{Z} \to \mathbb{Z}_3, \, x \mapsto x^2 \, (\text{mod} \, 3) \). (iv) \( \phi : \mathbb{Z} \to \mathbb{Z}_3, \, x \mapsto x^3 \, (\text{mod} \, 3) \). **Options:** A. All B. (i), (iii), (iv) C. (ii), (iii), (iv) D. (i), (ii), (iv) E. None
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