Exercise · ZW-GROUP-ISOMORPHISM-004 · ★★☆ Intermediate
Are \(\mathbb{Z}_4\) and \(\mathbb{Z}_2\times\mathbb{Z}_2\) isomorphic?
Choose one option.
Show answer
- C. No, because \(\mathbb{Z}_4\) has an element of order 4 and the other does not
Explanation
An isomorphism preserves element orders. In \(\mathbb{Z}_4\) the element \(1\) has order \(4\); in \(\mathbb{Z}_2\times\mathbb{Z}_2\) every element has order at most \(2\). Both being abelian of order \(4\) is not enough.
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
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- 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?