I am just reading Allufi chapter 0. I have a specific question in regards to a comment that the book made.
"By Proposition 6.9, every nontrivial subgroup of $\mathbb{Z}$ is in fact iso-morphic to $\mathbb{Z}$. Putting this a little strangely, it says that every subgroup of the free group on one generator is free."
I am not sure I understand this. Is it saying that subgroup of free groups on one generator is isomorphic to a subgroup of free groups of one generator ? I am guessing I am right in my understanding, because the second comment it is saying beware that free groups on two generators contain subgroups isomorphic to free group of arbitrary generators.
Yes, you understood it right. The only free group of one generator is $\mathbb{Z}$ up to isomorphism. So its non trivial subgroups are isomorphic to it. With free groups of more than one generator it becomes much more complicated.