Prove that every subgroup of an infinite cyclic group is characteristic

Question

Prove that every subgroup of an infinite cyclic group is characteristic.

I know that every infinite cyclic group is isomorphic to $\Bbb Z$, and any automorphism on $\Bbb Z$ is of the form $\alpha(n) = n$ or $\alpha(n) = -n$. That means that if $f$ is an isomorphism from $\Bbb Z$ to some other group $G$, the isomorphism is determined by $f(1)$. But from here I can't figure out how to show that it's characteristic.

Answer 1

Every subgroup $H$ of $\mathbb{Z}$ are is in the form $H=n\mathbb{Z}$ for some $n \in \mathbb{N}$. Those are clearly stable with respect to automorphisms of $\mathbb{Z}$, which, as you said, consist of the identity automorphism and the automorphism which sends $m$ to $-m$, for every $m \in \mathbb{Z}$.

Answer 2

For each $n \in \mathbb N^*$, there is exactly one subgroup of $\mathbb Z$ of index $n$, namely $n\mathbb Z$. Therefore, it must be fixed by every automorphism.