I'm struggling with the following question:
Prove that if $G$ is a subgroup of $\langle Z; +\rangle$ then $G=\{n·d\mid n∈\mathbb{Z}\}$ for some $d \in \mathbb{Z}$.
I feel this is too generic, I don't even know how to start :/ Many thanks for your help!
Lets make the comments into a proof. If $G\neq\{0\}$ then pick $d$ to be the smallest positive element of $G$.
To see that $G=\{n\cdot d\mid n\in\mathbb{Z}\}$, suppose there exists some $k\in G$ such that $k\neq n\cdot d$ for any $n\in\mathbb{Z}$. By the division algorithm, there exists some remainder $d>r>0$ such that $k=m\cdot d+r$. As $G$ is a subgroup, it follows that $r=k-m\cdot d\in G$. This is a contradiction, as $d$ is the smallest positive element of $G$ but $d>r>0$. Hence, there is no such $k\in G$, so $G$ has the required form.