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?
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?
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?
Show Hint
For problems involving quotient groups, analyze the properties of divisibility and index within the group structure.
Updated On: Feb 1, 2025
- Both I and II are TRUE
- I is TRUE and II is FALSE
- I is FALSE and II is TRUE
- Both I and II are FALSE
Show Solution
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The Correct Option is D
Solution and Explanation
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 \( {(4)} \).
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