Question

Consider the following statements:
1. There exists a proper subgroup \( G \) of \( ({Q}, +) \) such that \( {Q}/G \) is a finite group.
2. There exists a subgroup \( G \) of \( ({Q}, +) \) such that \( {Q}/G \) is isomorphic to \( ({Z}, +) \).
Which one of the following is correct?

• A. Both I and II are TRUE
• B. I is TRUE and II is FALSE
• C. I is FALSE and II is TRUE
• D. Both I and II are FALSE

Hint:

For problems involving quotient groups, analyze the properties of divisibility and index within the group structure.

Answer 1

The Correct Option is D

Step 1: Analyzing statement I. A proper subgroup \( G \) of \( ({Q}, +) \) cannot make \( {Q}/G \) a finite group because \( ({Q}, +) \) is infinitely divisible and does not contain finite index subgroups. Step 2: Analyzing statement II. It is impossible to construct a subgroup \( G \) of \( ({Q}, +) \) such that \( {Q}/G \) is isomorphic to \( ({Z}, +) \) because \( ({Q}, +) \) is divisible, whereas \( ({Z}, +) \) is not. Step 3: Conclusion. Both statements are false. The correct answer is \( {(4)} \).