Exercise · ZW-GROUP-ISOMORPHISM-001 · ★☆☆ Basic
Which of these groups is isomorphic to \((\mathbb{Z}_6,+)\)?
Choose one option.
Show answer
- A. \(\mathbb{Z}_2\times\mathbb{Z}_3\)
Explanation
Since \(\gcd(2,3)=1\), \(\mathbb{Z}_2\times\mathbb{Z}_3\) is cyclic of order \(6\), like \(\mathbb{Z}_6\). \(S_3\) is not abelian and the other two have orders \(4\) and \(9\).
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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