For each $\displaystyle a \ne 0$ and $\displaystyle b$ in $\displaystyle GF(7) = \bold{Z}_7$ define a function $\displaystyle \mathit{F}_{a,b}:\bold{Z}_7\rightarrow \bold{Z}_7$ by

$\displaystyle \mathit{F}_{a,b}(x) = ax + b$

The set of all such function $\displaystyle \mathit{F}_{a,b}$ forms a group $\displaystyle g$ with the group multiplication given by the compostion of the functions.(Not need to verify $\displaystyle G$ is a group)

a. Determine $\displaystyle |G|$ and find $\displaystyle \mathit{F}_{a,b}\circ \mathit{F}_{c,d} = \mathit{F}_{a,b}(\mathit{F}_{c,d}(x))$. Hence show that $\displaystyle G$ is non-abelian.

b. Show that $\displaystyle \mathit{f}:G\rightarrow(\bold{Z}_7\setminus {0},\cdot )$,given by $\displaystyle \mathit{f}(\mathit{F}_{a,b})=a$, is a homomorphism.

I'm OK with the $\displaystyle |G|$ part but I forget how to show $\displaystyle \mathit{F}_{a,b}\circ \mathit{F}_{c,d} = \mathit{F}_{a,b}(\mathit{F}_{c,d}(x))$.

Also for part b, I don't really get it.

Thank you for help