Question

Let \( H \) be the quotient group \( \mathbb{Q} / \mathbb{Z} \). Consider the following statements.
I. Every cyclic subgroup of \( H \) is finite.
II. Every finite cyclic group is isomorphic to a subgroup of \( H \).
Which one of the following holds?

• A. I is TRUE but II is FALSE
• B. II is TRUE but I is FALSE
• C. both I and II are TRUE
• D. neither I nor II is TRUE

Hint:

In quotient groups like \( \mathbb{Q} / \mathbb{Z} \), cyclic subgroups are finite, and every finite cyclic group can be isomorphic to a subgroup of \( H \).

Answer 1

The Correct Option is C

Step 1: Analyzing statement I.
Every cyclic subgroup of \( H \) is finite. Since \( H = \mathbb{Q} / \mathbb{Z} \), any cyclic subgroup generated by a rational number has finite order, as it is a quotient of the integers. Thus, statement I is TRUE.

Step 2: Analyzing statement II.
Any finite cyclic group can be embedded in \( \mathbb{Q} / \mathbb{Z} \) because every finite cyclic group is isomorphic to some subgroup of \( \mathbb{Q} / \mathbb{Z} \), which is a direct sum of cyclic groups. Thus, statement II is TRUE.

Step 3: Conclusion.
The correct answer is (C) both I and II are TRUE.