Exercise · ZW-GROUP-ISOMORPHISM-008 · ★★☆ Intermediate
How many isomorphisms \((\mathbb{Z}_{10},+)\to(\mathbb{Z}_{10},+)\) are there?
Choose one option.
Show answer
- C. 4
Explanation
Each one sends \(1\) to a generator, i.e. to some \(a\) with \(\gcd(a,10)=1\): \(a\in\{1,3,7,9\}\). That gives \(\varphi(10)=4\).
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-009 Which of these groups of order 12 is isomorphic to ℤ_12?