Exercise · ZW-GROUP-ISOMORPHISM-007 · ★★☆ Intermediate
Why are \(\mathbb{Z}_6\) and \(S_3\) not isomorphic, even though both have order 6?
Choose one option.
Show answer
- B. Because \(\mathbb{Z}_6\) is abelian and \(S_3\) is not
Explanation
Being abelian is preserved by isomorphisms. In \(\mathbb{Z}_6\) every pair commutes; in \(S_3\) there are permutations \(\sigma,\tau\) with \(\sigma\tau\ne\tau\sigma\).
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-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-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?