Chapter 3.3, Problem 28E

For each $n\in \mathbb{Z}$, define $f_n:\mathbb{Z}\to \mathbb{Z}$ by $f_n(x)=x+n$ for $x\in \mathbb{Z}$.

a. Show that $f_n$ is an element of $S(\mathbb{Z})$.

b. Let $H=\left\{f_n\in S(\mathbb{Z})|f_n(x)=x+nforeachn\in \mathbb{Z}\right\}$.Prove that $H$ is a subgroup of $S(\mathbb{Z})$ under mapping composition.

c. Prove that $H$ is abelian, even though $S(\mathbb{Z})$ is not.