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?

• A. $\mathbb{Q}/\mathbb{Z}$ contains a subgroup isomorphic to $(\mathbb{Z}, +)$.
• B. There is exactly one group homomorphism from $\mathbb{Q}/\mathbb{Z}$ to $(\mathbb{Q}, +)$.
• C. For all $n \in \mathbb{N}$, there exists $g \in \mathbb{Q}/\mathbb{Z}$ such that the order of $g$ is $n$.
• D. $\mathbb{Q}/\mathbb{Z}$ is not a cyclic group.

Answer 1

Answer: (A)

$\mathbb{Q}/\mathbb{Z}$ contains a subgroup isomorphic to $(\mathbb{Z}, +)$.

Answer 2

The correct option is (A): $\mathbb{Q}/\mathbb{Z}$ contains a subgroup isomorphic to $(\mathbb{Z}, +)$.