Show that $(\mathbb{Q}, +)$ and $(\mathbb{R}, +)$ are not cyclic groups.

Question

How do I show that $(\mathbb{Q}, +)$ and $(\mathbb{R}, +)$ are not cyclic groups? (my teacher suggested solving this by using contradction, but I don't know where to start)

Answer 1

Hint: Assume that $\mathbb{Q}$ under addition is cyclic. Then there is a rational numeber $p/q$ such that $\{np/q: n\in \mathbb{Z}\} = \mathbb{Q}$. Now just find one rational number that is not in this set.

For $\mathbb{R}$ recall that a subgroup of a cyclic group is cyclic. So since $\mathbb{Q}$ is a subgroup of $\mathbb{R}$ ...

Answer 2

Hint: Suppose $g$ is a generator. Note that $g$ cannot be $0$ (why?). Consider $\frac12g.$

Answer 3

Suppose $(\mathbb{Q},+)$ is cylclic. Then $\mathbb{Q}=\langle q\rangle$ for some $q\in\mathbb{Q}$ and clearly $q\neq 0$. Now since $\frac q2$ is rational, $nq=\frac q2$ for some $n\in\mathbb{Z}$ which gives $n=\frac12$ which is a contradiction.