For a positive integer $ n $, let $ U(n) = \{ r \in \mathbb{Z}_n : \gcd(r, n) = 1 \} $ be the group under multiplication modulo $ n $. Then, which one of the following statements is TRUE?

Question

cdquestions Admin · Accepted Answer

We begin by finding the elements of the groups $ U(5) $, $ U(8) $, $ U(10) $, and $ U(12) $: - $ U(5) = \{ 1, 2, 3, 4 \} $ because the numbers less than 5 and relatively prime to 5 are 1, 2, 3, and 4. The order of $ U(5) $ is 4. - $ U(8) = \{ 1, 3, 5, 7 \} $ because the numbers less than 8 and relatively prime to 8 are 1, 3, 5, and 7. The order of $ U(8) $ is 4. - $ U(10) = \{ 1, 3, 7, 9 \} $ because the numbers less than 10 and relatively prime to 10 are 1, 3, 7, and 9. The order of $ U(10) $ is 4. - $ U(12) = \{ 1, 5, 7, 11 \} $ because the numbers less than 12 and relatively prime to 12 are 1, 5, 7, and 11. The order of $ U(12) $ is 4. Next, observe that $ U(8) $ and $ U(12) $ both have the same number of elements (4), and they both have identical group structures. Both are cyclic groups of order 4, and they are isomorphic to each other. Therefore, the correct answer is (D).

Answer

$ U(5) $ is isomorphic to $ U(8) $

Answer

$ U(10) $ is isomorphic to $ U(12) $

Answer

$ U(8) $ is isomorphic to $ U(10) $

Answer

$ U(8) $ is isomorphic to $ U(12) $

For a positive integer \( n \), let \( U(n) = \{ r \in \mathbb{Z}_n : \gcd(r, n) = 1 \} \) be the group under multiplication modulo \( n \). Then, which one of the following statements is TRUE?

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The Correct Option is D

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