Question:
For \( n \in \mathbb{N} \), consider the set \( U(n) = \{ x \in \mathbb{Z}_n : \gcd(x, n) = 1 \} \) as a group under multiplication modulo \( n \).
Then, which of the following is/are TRUE?
For \( n \in \mathbb{N} \), consider the set \( U(n) = \{ x \in \mathbb{Z}_n : \gcd(x, n) = 1 \} \) as a group under multiplication modulo \( n \).
Then, which of the following is/are TRUE?
Show Hint
To determine if a group is cyclic, check if there exists an element that generates all the other elements of the group through repeated application of the group operation.
Updated On: Sep 6, 2025
- \( U(8) \) is a cyclic group.
- \( U(5) \) is a cyclic group.
- \( U(12) \) is a cyclic group.
- \( U(9) \) is a cyclic group.
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The Correct Option is B
Solution and Explanation
Step 1: Check if \( U(5) \) is cyclic.
The set \( U(5) = \{1, 2, 3, 4\} \) consists of numbers less than 5 that are coprime to 5. We check if there exists an element that can generate all elements of the set under multiplication modulo 5: - \( 1^k \equiv 1 \pmod{5} \) - \( 2^1 \equiv 2 \pmod{5}, \, 2^2 \equiv 4 \pmod{5}, \, 2^3 \equiv 3 \pmod{5}, \, 2^4 \equiv 1 \pmod{5} \) - Hence, \( 2 \) is a generator, and \( U(5) \) is cyclic. Step 2: Check \( U(8) \), \( U(12) \), and \( U(9) \).
We find that \( U(8) \) and \( U(12) \) are not cyclic based on their structure, and \( U(9) \) is not cyclic either. Final Answer: \[ \boxed{U(5) \text{ is a cyclic group.}} \]
The set \( U(5) = \{1, 2, 3, 4\} \) consists of numbers less than 5 that are coprime to 5. We check if there exists an element that can generate all elements of the set under multiplication modulo 5: - \( 1^k \equiv 1 \pmod{5} \) - \( 2^1 \equiv 2 \pmod{5}, \, 2^2 \equiv 4 \pmod{5}, \, 2^3 \equiv 3 \pmod{5}, \, 2^4 \equiv 1 \pmod{5} \) - Hence, \( 2 \) is a generator, and \( U(5) \) is cyclic. Step 2: Check \( U(8) \), \( U(12) \), and \( U(9) \).
We find that \( U(8) \) and \( U(12) \) are not cyclic based on their structure, and \( U(9) \) is not cyclic either. Final Answer: \[ \boxed{U(5) \text{ is a cyclic group.}} \]
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