Exercise · ZW-GROUP-ISOMORPHISM-005 · ★☆☆ Basic
The function \(x\mapsto e^x\) is an isomorphism from \((\mathbb{R},+)\) onto \((\mathbb{R}_{>0},\cdot)\). What is its inverse isomorphism?
Choose one option.
Show answer
- B. \(y\mapsto \ln y\)
Explanation
The inverse must undo \(e^x\): \(\ln(e^x)=x\) and \(e^{\ln y}=y\). Moreover \(\ln(yz)=\ln y+\ln z\), so it preserves the operation.
Review the concept: Group isomorphism
A group isomorphism is a bijective homomorphism \(\varphi\) from a group \(G\) to a group \(H\); then \(G\) and \(H\) are isomorphic.
\[G\cong H\]
More practice problems: group isomorphism
- ZW-GROUP-ISOMORPHISM-001 Which of these groups is isomorphic to (ℤ_6,+)?
- ZW-GROUP-ISOMORPHISM-002 If G and H are isomorphic, which of these statements can be false?
- ZW-GROUP-ISOMORPHISM-003 What is the largest order of an element of ℤ_2×ℤ_4?
- ZW-GROUP-ISOMORPHISM-004 Are ℤ_4 and ℤ_2×ℤ_2 isomorphic?
- ZW-GROUP-ISOMORPHISM-006 How many isomorphisms (ℤ_6,+)→(ℤ_6,+) are there?
- ZW-GROUP-ISOMORPHISM-007 Why are ℤ_6 and S_3 not isomorphic, even though both have order 6?
- ZW-GROUP-ISOMORPHISM-008 How many isomorphisms (ℤ_10,+)→(ℤ_10,+) are there?
- ZW-GROUP-ISOMORPHISM-009 Which of these groups of order 12 is isomorphic to ℤ_12?