Exercise · ZW-GROUP-ISOMORPHISM-010 · ★★★ Advanced
How many groups of order 4 exist up to isomorphism?
Choose one option.
Show answer
- A. 2
Explanation
Every group of order \(4\) either has an element of order \(4\) (and is \(\mathbb{Z}_4\)) or all its non-identity elements have order \(2\) (and it is \(\mathbb{Z}_2\times\mathbb{Z}_2\)). These two are not isomorphic, so there are exactly \(2\).
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-008 How many isomorphisms (ℤ_10,+)→(ℤ_10,+) are there?