Prove that for the additive group $(\mathbb{Z}, +)$ of integers every subgroup is of the form $k\mathbb{Z}$.
Here is the proof I wrote, and I knew it was off, so I sent it to my professor for some help. His reply: "Sorry, but you missed the point. You needed to show that any subgroup of $\mathbb{Z}$ is of the form $k\mathbb{Z}$ for some $k$. Keep on trying."
Which is less than helpful. This is a problem off a practice final.
Can someone give me a hint to get me started?
edit: here's my new attempt. LINK Is this enough? Or is more needed?
Best Answer
What you proved is that $k\mathbb{Z}$ is a subgroup for any $k$. But to prove the statement given to you, your proof should begin: "Let $H$ be a subgroup of $\mathbb{Z}$" and conclude with "Therefore $H = k\mathbb{Z}$ for some $k \in \mathbb{Z}$."
If $H$ is a subgroup of $\mathbb{Z}$, try looking at the smallest (in absolute value) element of $H$.