Exercise · ZW-GROUP-ISOMORPHISM-009 · ★★★ Advanced
Which of these groups of order 12 is isomorphic to \(\mathbb{Z}_{12}\)?
Choose one option.
Show answer
- A. \(\mathbb{Z}_3\times\mathbb{Z}_4\)
Explanation
Since \(\gcd(3,4)=1\), \(\mathbb{Z}_3\times\mathbb{Z}_4\) has an element of order \(12\) and is cyclic. In \(\mathbb{Z}_2\times\mathbb{Z}_6\) and \(\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_3\) the largest order is \(6\), and \(A_4\) is not abelian.
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.
\[G\cong H\]
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