Question

For a positive integer n, let U(n) = {\(\bar{r}\) ∈ ℤ_n ∶ gcd(r, n) = 1} be the group under multiplication modulo n. Then, which one of the following statements is TRUE ?

• A. U(5) is isomorphic to U(8)
• B. U(10) is isomorphic to U(12)
• C. U(8) is isomorphic to U(10)
• D. U(8) is isomorphic to U(12)

Answer 1

The Correct Option is D

To determine which groups \( U(n) \) are isomorphic, we first need to understand the structure of these groups.

Given a positive integer \( n \), the group \( U(n) \) is the set of integers less than \( n \) that are relatively prime to \( n \), with multiplication modulo \( n \) as the group operation.

The order (or size) of the group \( U(n) \) is given by Euler's totient function \( \phi(n) \), which counts the number of integers up to \( n \) that are coprime to \( n \).

Let's compute the orders of the given groups:

1. For \( U(5) \): Since 5 is prime, \( \phi(5) = 5 - 1 = 4 \).
2. For \( U(8) \): \( \phi(8) = 8 \times \left(1 - \frac{1}{2}\right) = 4 \). (8 is a power of 2)
3. For \( U(10) \): \( \phi(10) = 10 \times \left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{5}\right) = 4 \).
4. For \( U(12) \): \( \phi(12) = 12 \times \left(1 - \frac{1}{2}\right) \times \left(1 - \frac{1}{3}\right) = 4 \).

We find that \( U(5), U(8), U(10), \) and \( U(12) \) each have 4 elements. Groups with the same number of elements can be isomorphic if they have the same structure.

Let's analyze each pair to find out which one is actually isomorphic:

• \( U(5) \) and \( U(8) \): Being different cyclic structures (one is generated by 2 modulo 5, another by 3 and 5 modulo 8), they cannot be isomorphic.
• \( U(10) \) and \( U(12) \): Their respective structures differ because one involves coprimes to a product of 2 and 5, another with 2 and 3. They have different element behaviors.
• \( U(8) \) and \( U(10) \): Similarly, different prime factor components lead to different multiplication structures.
• \( U(8) \) and \( U(12) \): Both being able to generate cyclic groups of order 4 (e.g. \( \{1, 3, 5, 7\}\) for \( U(8)\) and subgroup elements like \( \{1, 5, 7, 11\} \) for \( U(12)\)) tend to align closely in structure, potentially allowing for isomorphism.

After considering the group structures, \( U(8) \) is isomorphic to \( U(12) \) because they both can form cyclic groups of order 4 sharing similar component interactions under multiplication.

Conclusion: The correct statement is: U(8) is isomorphic to U(12).