Question:

Consider the group (Q,+) (\mathbb{Q}, +) and its subgroup (Z,+) (\mathbb{Z}, +) .
For the quotient group Q/Z \mathbb{Q}/\mathbb{Z} , which one of the following is FALSE?

Updated On: Oct 1, 2024
  • Q/Z \mathbb{Q}/\mathbb{Z} contains a subgroup isomorphic to (Z,+) (\mathbb{Z}, +) .
  • There is exactly one group homomorphism from Q/Z \mathbb{Q}/\mathbb{Z} to (Q,+) (\mathbb{Q}, +) .
  • For all nN n \in \mathbb{N} , there exists gQ/Z g \in \mathbb{Q}/\mathbb{Z} such that the order of g g is n n .
  • Q/Z \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): Q/Z \mathbb{Q}/\mathbb{Z} contains a subgroup isomorphic to (Z,+) (\mathbb{Z}, +) .
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