find a generator for the group $G =\{ f(x) = x+n\mid n\in \Bbb Z \}$ with the group operation being composition.

Question

Another question from 'A book of Abstract Algebra' by Pinter.

For each $n\in \Bbb Z$ define $f_n = x+n$. Then $f_n\in S_{\Bbb R}$, the symmetric set on $\Bbb R$. The group operation being composition. Now $f_n \circ f_m =f_{m+n}$ and $f^{-1}_n=f_{-n}$

Let $G=\{ f_n \mid n\in \Bbb Z \}$ such that $G$ is a subgroup of $S_{\Bbb R}$. Show that $G$ is cyclic by finding a generator.

I cannot seem to find the right generator. Using $f_1$ as a generator produces all $f_n$ for $n = 1,2,3,\ldots$ but then the identity $f_0$ is not included and the inverses are missing. Similarly for $f_{-1}$. What am I missing? I just need to find the generator but I cannot seem to find it. Thanks in advance.

Answer 1

Hint: Prove $G \cong \mathbb Z$.

Answer 2

Here $f_0$ is the identity map and is included by hypothesis, so your generator $f_1$ is correct. Here $f_{-1}$ is not missing: it is the inverse of $f_1$!