Let $f \in Map(\mathbb{Z})$ be given by $f(n) = n+1$ for every $n \in \mathbb{Z}$. Find all elements in $Map(\mathbb{Z})$ that commute with $f$.

Question

Let $f \in Map(\mathbb{Z})$ be given by $f(n) = n+1$ for every $n \in \mathbb{Z}$. Find all elements in $Map(\mathbb{Z})$ that commute with $f$.

I understood that I need to show $f^m$ $\forall m \in \mathbb{Z}$ commute with $f$. For the positive integer $m$ it is easy but what can I do for negative integers? It would mean to find the inverse I guess. Also how do I show that any other element of $Map(\mathbb{Z})$ doesn't commute with $f$?

Please some help..!!

Edit : I think for the negative index $m \in \mathbb{Z}$, $f^{-m}(n) = n -m$, it can also be shown easily that for all integer $m$, $f$ commutes with $f^m$.

Answer 1

Let $g$ commute with $f$, then $g(f(n))=f(g(n))$, which implies $g(n+1)=g(n)+1$, so we only need to know one value of $g$ to determine the function. If $g(0)=m$ then $g(n)=m+n$ and so the set of functions commuting with $f$ is: $$\lbrace g\in Map(\mathbb{Z})\mid g(n)=n+m \forall n\in\mathbb{Z} \text{ for some } m\in\mathbb{Z}\rbrace$$