Results 1 to 2 of 2

Thread: Homomorphism and group problem

  1. #1
    Junior Member
    Joined
    Mar 2008
    Posts
    33

    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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    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 $\displaystyle f(F_{a,b}\circ F_{c,d}) = f(F_{a,b})f(F_{c,d})$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Group homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Nov 11th 2011, 04:45 PM
  2. group homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Nov 22nd 2009, 01:18 AM
  3. Group homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Sep 24th 2009, 04:00 PM
  4. group homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Feb 22nd 2009, 09:54 AM
  5. Homomorphism of P group
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Nov 18th 2008, 08:07 PM

Search Tags


/mathhelpforum @mathhelpforum