Exercise · ZW-GROUP-ISOMORPHISM-003 · ★★☆ Intermediate
What is the largest order of an element of \(\mathbb{Z}_2\times\mathbb{Z}_4\)?
Choose one option.
Show answer
- A. 4
Explanation
A pair \((a,b)\) has order the least common multiple of the orders of \(a\) and \(b\), at most \(\operatorname{lcm}(2,4)=4\). There is no element of order \(8\), so \(\mathbb{Z}_2\times\mathbb{Z}_4\not\cong\mathbb{Z}_8\).
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-005 The function xmapsto e^x is an isomorphism from (ℝ,+) onto (ℝ_>0,·). What is its inverse isomorphis…
- 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?