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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).

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).

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) \( \phi: \mathbb{Z}_2 \rightarrow \mathbb{Z}_2, x \mapsto x^2 \pmod{2} \). (ii) \( \phi: \mathbb{Z} \rightarrow \mathbb{Z}_2, x \mapsto x^2 \pmod{2} \). (iii) \( \phi: \mathbb{Z} \rightarrow \mathbb{Z}_3, x \mapsto x^2 \pmod{3} \). (iv) \( \phi: \mathbb{Z} \rightarrow \mathbb{Z}_3, x \mapsto x^3 \pmod{3} \). **Options:** A) All B) (i), (iii), (iv) C) (ii), (iii), (iv) D) (i), (ii), (iv) E) None **Explanation:** To determine which mappings are group homomorphisms, verify that the functions preserve the operation of addition under modulo given in the mappings.
Transcribed Image Text:**Question:** Which of the following are group homomorphisms? (i) \( \phi: \mathbb{Z}_2 \rightarrow \mathbb{Z}_2, x \mapsto x^2 \pmod{2} \). (ii) \( \phi: \mathbb{Z} \rightarrow \mathbb{Z}_2, x \mapsto x^2 \pmod{2} \). (iii) \( \phi: \mathbb{Z} \rightarrow \mathbb{Z}_3, x \mapsto x^2 \pmod{3} \). (iv) \( \phi: \mathbb{Z} \rightarrow \mathbb{Z}_3, x \mapsto x^3 \pmod{3} \). **Options:** A) All B) (i), (iii), (iv) C) (ii), (iii), (iv) D) (i), (ii), (iv) E) None **Explanation:** To determine which mappings are group homomorphisms, verify that the functions preserve the operation of addition under modulo given in the mappings.
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