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Math Help - Homomorphism and group problem

  1. #1
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    Homomorphism and group problem

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

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

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

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

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


    I'm OK with the |G| part but I forget how to show \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
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  2. #2
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    a) F_{a,b}\circ F_{c,d} = F_{a,b}(cx+d) = a(cx+d)+b = acx + ad+b.

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