Exercise · ZW-GROUP-HOMOMORPHISM-005 · ★☆☆ Basic
If \(\varphi\colon G\to H\) is a group homomorphism and \(e_G\) is the identity of \(G\), what is \(\varphi(e_G)\)?
Choose one option.
Show answer
- A. The identity \(e_H\) of \(H\)
Explanation
From \(\varphi(e_G)=\varphi(e_Ge_G)=\varphi(e_G)\varphi(e_G)\), cancelling one factor leaves \(\varphi(e_G)=e_H\), always.
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-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?