Exercise · ZW-GROUP-ISOMORPHISM-002 · ★☆☆ Basic
If \(G\) and \(H\) are isomorphic, which of these statements can be false?
Choose one option.
Show answer
- B. G and H consist of exactly the same objects as elements
Explanation
An isomorphism preserves only structure, not what the elements are: \(\mathbb{Z}_4\) and the fourth roots of unity are isomorphic yet consist of different objects. The other three properties are always preserved.
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-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-005 The function xmapsto e^x is an isomorphism from (ℝ,+) onto (ℝ_>0,·). What is its inverse isomorphis…
- 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?