Question:
Consider the group \( (\mathbb{Q}, +) \) and its subgroup \( (\mathbb{Z}, +) \).
For the quotient group \( \mathbb{Q}/\mathbb{Z} \), which one of the following is FALSE?
Consider the group \( (\mathbb{Q}, +) \) and its subgroup \( (\mathbb{Z}, +) \).
For the quotient group \( \mathbb{Q}/\mathbb{Z} \), which one of the following is FALSE?
For the quotient group \( \mathbb{Q}/\mathbb{Z} \), which one of the following is FALSE?
Updated On: Oct 1, 2024
- \( \mathbb{Q}/\mathbb{Z} \) contains a subgroup isomorphic to \( (\mathbb{Z}, +) \).
- There is exactly one group homomorphism from \( \mathbb{Q}/\mathbb{Z} \) to \( (\mathbb{Q}, +) \).
- For all \( n \in \mathbb{N} \), there exists \( g \in \mathbb{Q}/\mathbb{Z} \) such that the order of \( g \) is \( n \).
- \( \mathbb{Q}/\mathbb{Z} \) is not a cyclic group.
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The Correct Option is A
Solution and Explanation
The correct option is (A): \( \mathbb{Q}/\mathbb{Z} \) contains a subgroup isomorphic to \( (\mathbb{Z}, +) \).
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