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Well-defined Homomorphism

  1. Sep 20, 2016 #1
    1. The problem statement, all variables and given/known data
    Determine the integers ##k## for which ##f_k : \mathbb{Z}/ 48 \mathbb{Z}## with ##f_k (\overline{1}) = x^k## extends to a well-defined homomorphism.

    2. Relevant equations


    3. The attempt at a solution
    My claim is that ##f_k## extends to a well-defined homomorphism iff ##f(\overline{b}) = x^{bk}## for every ##\overline{b} \in \mathbb{Z}/48\mathbb{Z}## and ##k## is such that ##36 divides ##48k# (which is equivalent to ##3## dividing ##k##). I was able to prove that if ##f_k(\overline{b})=x^{kb}## and ##k## is such that ##36## divides ##48k##, then ##f## is a well-defined hommorphism. However, I am having difficulty with the other direction.

    Suppose that ##f## is a well-defined homomorphism. Then

    ##
    f_k(\overline{b}) = f_k(\overline{1} + \dots + \overline{1}) = f_k(\overline{1}) \dots f_k(\overline{1}) = x^k \dots x^k = x^{kz}##

    Now we want to show ##k## is such that ##36## divides ##48k##. Suppose the contrary, and suppose ##\overline{b} = \overline{b'}##, which implies ##b' = b + 48m## Then by the well-defined property,

    ##f_k(\overline{b}) = f_k(\overline{b'})##

    ##x^{bk} = x^{b'k}##

    ##x^{bk} = x^{(b+48m)k}##

    ##x^{bk} = x^{bk} (x^{48k})^m##

    ##e = (x^{48k})^m##

    This is where I get stuck...
     
  2. jcsd
  3. Sep 20, 2016 #2
    Note, ##x## is the generator of ##Z_{36}##.
     
  4. Sep 20, 2016 #3

    pasmith

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    Homework Helper

    You haven't specified the codomain of [itex]f_k[/itex]. Is it [itex]\mathbb{Z} / 36 \mathbb{Z}[/itex] as your attempt suggests?
     
  5. Sep 20, 2016 #4
    Yes, I am sorry. It is actually ##Z_{36}##, which is of course isomorphic to it.
     
  6. Sep 20, 2016 #5

    pasmith

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    You don't need to prove two directions separately; you can do both at once.

    [itex]f_k[/itex] is well-defined if and only if [itex]f_k(\overline b) = f_k(\overline c)[/itex] whenever [itex]b[/itex] and [itex]c[/itex] are in the same equivalence class. Thus you need [itex]x^{bk} = x^{(b+ 48q)k} = x^{bk}x^{48qk}[/itex] for each integer [itex]q[/itex].

    Your hypothesis is that well-definedness of [itex]f_k[/itex] is governed by whether 36 divides 48k. So set [itex]48k = 36p + r[/itex] for integers [itex]p \in \mathbb{Z}[/itex] and [itex]r \in \{0, 1, \dots, 35\}[/itex]. For which values of [itex]r[/itex] can you satisfy the above condition for all [itex]q[/itex]?
     
  7. Sep 20, 2016 #6
    What is ##q##? Should it be ##p##? The only ##r## that would satisfy the equation is ##r=0##, right?
     
    Last edited: Sep 20, 2016
  8. Sep 20, 2016 #7
    Sorry I misread what you wrote. I was able to work problem and it agrees with what you suggested. THANKS!
     
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