Exercise · ZW-GROUP-HOMOMORPHISM-006 · ★★☆ Intermediate
Let \(\varphi\colon(\mathbb{Z},+)\to(\mathbb{Z}_7,+)\) be the homomorphism \(\varphi(n)=3n \bmod 7\). What is \(\varphi(-2)\)?
Choose one option.
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- B. 1
Explanation
It is the inverse of \(\varphi(2)=6\) in \(\mathbb{Z}_7\), namely \(-6\equiv1\). Directly: \(3\cdot(-2)=-6\equiv1\pmod 7\). The value \(6\) is \(\varphi(2)\), not \(\varphi(-2)\).
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-001 Let φcolon(ℤ,+)→(ℤ_5,+) be given by φ(n)=n bmod 5, a homomorphism. What is φ(12)?
- 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-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?