Question

Let \( \mathbb{R}/\mathbb{Z} \) denote the quotient group, where \( \mathbb{Z} \) is considered as a subgroup of the additive group of real numbers \( \mathbb{R} \). Let \( m \) denote the number of injective (one-one) group homomorphisms from \( \mathbb{Z}_3 \) to \( \mathbb{R}/\mathbb{Z} \) and \( n \) denote the number of group homomorphisms from \( \mathbb{R}/\mathbb{Z} \) to \( \mathbb{Z}_3 \). Then, which one of the following is TRUE?

• A. \( m = 2 \) and \( n = 1 \)
• B. \( m = 3 \) and \( n = 3 \)
• C. \( m = 2 \) and \( n = 3 \)
• D. \( m = 1 \) and \( n = 1 \)

Hint:

To determine the number of injective homomorphisms, check the structure of the groups involved and the possible mappings. A quotient group may have fewer homomorphisms due to its structure.

Answer 1

The Correct Option is A

Step 1: Analyze the injective homomorphisms from \( \mathbb{Z}_3 \) to \( \mathbb{R}/\mathbb{Z} \).
An injective homomorphism must map each element of \( \mathbb{Z}_3 \) to a distinct element in \( \mathbb{R}/\mathbb{Z} \). There are 2 possible such homomorphisms. Step 2: Analyze the homomorphisms from \( \mathbb{R}/\mathbb{Z} \) to \( \mathbb{Z}_3 \).
Since \( \mathbb{R}/\mathbb{Z} \) is a quotient group, there is only one nontrivial homomorphism from it to \( \mathbb{Z}_3 \), so \( n = 1 \). Final Answer: \[ \boxed{m = 2 \text{ and } n = 1} \]