Exercise · ZW-GROUP-HOMOMORPHISM-010 · ★★★ Advanced
If \(\varphi\colon G\to H\) and \(\psi\colon H\to K\) are homomorphisms, which function is certainly a homomorphism?
Choose one option.
Show answer
- C. \(\psi\circ\varphi\colon G\to K\)
Explanation
Composition preserves the operation: \(\psi(\varphi(ab))=\psi(\varphi(a)\varphi(b))=\psi(\varphi(a))\psi(\varphi(b))\). The others may not exist, since \(\varphi\) or \(\psi\) need not be invertible.
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-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?