Are there any non-trivial automorphisms on the Natural Numbers under addition?

Question

I'm curious about whether or not there is an automorphism on $(\mathbb{N};+)$ that isn't the identity. I suspect there isn't, but I'm not quite sure how to prove it.

Answer 1

If $f:\mathbb{N}\to\mathbb{N}$ is any monoid-homomorphism, let $a=f(1)$. Then $f(2)=f(1+1)=f(1)+f(1)=2a$, $f(3)=f(2+1)=f(2)+f(1)=2a+a=3a$, and so on. More precisely, we can prove by induction that for each $n\in\mathbb{N}$, $f(n)=an$ (it is a good exercise to work out this induction if you don't immediately see how it works).

If $f$ is an automorphism, then it must be surjective, so every element of $\mathbb{N}$ is equal to $an$ for some $n$. That is, every natural number is divisible by $a$. This is only possible if $a=1$, and so $f(n)=an=n$ and $f$ is the identity.