Exercise · ZW-GROUP-HOMOMORPHISM-001 · ★☆☆ Basic
Let \(\varphi\colon(\mathbb{Z},+)\to(\mathbb{Z}_5,+)\) be given by \(\varphi(n)=n \bmod 5\), a homomorphism. What is \(\varphi(12)\)?
Choose one option.
Show answer
- B. 2
Explanation
Since \(12=2\cdot5+2\), the remainder is \(2\), so \(\varphi(12)=2\).
Review the concept: Group homomorphism
A group homomorphism is a function \(\varphi\) from a group \(G\) to a group \(H\) that respects the operation.
\[\varphi(ab)=\varphi(a)\,\varphi(b)\quad\text{for all } a,b\in G\]
More practice problems: group homomorphism
- ZW-GROUP-HOMOMORPHISM-002 Which of these functions (ℤ,+)→(ℤ,+) is a homomorphism?
- ZW-GROUP-HOMOMORPHISM-003 The function φcolon(ℝ,+)→(ℝ^*,·), φ(x)=e^x, is a homomorphism. What is φ(2)·φ(3) equal to?
- ZW-GROUP-HOMOMORPHISM-004 For the homomorphism detcolon GL_2(ℝ)→ℝ^* and matrices A,B, what is det(AB) equal to?
- ZW-GROUP-HOMOMORPHISM-005 If φcolon G→ H is a group homomorphism and e_G is the identity of G, what is φ(e_G)?
- ZW-GROUP-HOMOMORPHISM-006 Let φcolon(ℤ,+)→(ℤ_7,+) be the homomorphism φ(n)=3n bmod 7. What is φ(-2)?
- ZW-GROUP-HOMOMORPHISM-007 How many homomorphisms (ℤ_6,+)→(ℤ_6,+) exist?
- ZW-GROUP-HOMOMORPHISM-008 How many homomorphisms (ℤ_12,+)→(ℤ_8,+) exist?
- ZW-GROUP-HOMOMORPHISM-009 How many homomorphisms (ℤ_5,+)→(ℤ_3,+) exist?