Question

Consider the following statements. I. The group \((\mathbb{Q}, +)\) has no proper subgroup of finite index.
II. The group \((\mathbb{C} \setminus \{0\}, \cdot)\) has no proper subgroup of finite index.
Which one of the following statements is true?

• A. Both I and II are TRUE.
• B. I is TRUE but II is FALSE.
• C. II is TRUE but I is FALSE.
• D. Neither I nor II is TRUE.

Hint:

Divisible groups like \((\mathbb{Q}, +)\) typically lack proper finite index subgroups, but multiplicative groups of complex numbers can contain finite cyclic subgroups.

Answer 1

The Correct Option is A

Step 1: Analyze Statement I.
\((\mathbb{Q}, +)\) is a divisible group, meaning for every \(q \in \mathbb{Q}\) and integer \(n>0\), there exists \(r \in \mathbb{Q}\) such that \(nr = q\). Divisible groups have no proper subgroups of finite index. Hence, (I) is TRUE.
Step 2: Analyze Statement II.
\((\mathbb{C}\setminus\{0\}, \cdot)\) is also divisible, but it contains proper subgroups like the group of \(n^{\text{th}}\) roots of unity, which have finite index. Thus, (II) is FALSE.
Step 3: Conclusion.
Hence, the correct choice is (B).