# Homomorphism and group problem

• Mar 31st 2008, 10:17 PM
kleenex
Homomorphism and group problem
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
• Apr 1st 2008, 07:12 AM
ThePerfectHacker
a)$\displaystyle F_{a,b}\circ F_{c,d} = F_{a,b}(cx+d) = a(cx+d)+b = acx + ad+b$.

b)$\displaystyle G$ is the set of all these linear transformations that are homomorphisms of $\displaystyle \mathbb{Z}_7$. Let $\displaystyle x\in G$, then it has the form $\displaystyle x=F_{a,b}$, by contrustion. We define a mapping $\displaystyle f:G\mapsto (\mathbb{Z}_7\setminus \{ 0 \} )$ as $\displaystyle f(F_{a,b})=a$, the first coefficient. You need to show $\displaystyle f(F_{a,b}\circ F_{c,d}) = f(F_{a,b})f(F_{c,d})$.