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Proving that 'a' can be written as the nth power

  1. Oct 18, 2012 #1

    Zondrina

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    1. The problem statement, all variables and given/known data

    Let a belong to a group and |a| = m. If n is relatively prime to m, show that
    a can be written as the nth power of some element in the group.

    2. Relevant equations

    We know :
    |a| = m
    gcd(m,n) = 1

    We want to show :
    bn = a for some b in the group.

    3. The attempt at a solution

    Okay, I didn't really know where to start with this one, but I'll give it a try.

    We know the gcd can be written as a linear combination, that is :

    gcd(m,n) = 1 = ms + nt for some integers s and t.

    Now :
    1 = ms + nt
    a1 = ams + nt
    a = ams ant
    a = easant
    a = asant

    Here's where I get stuck. Any pointers?
     
  2. jcsd
  3. Oct 18, 2012 #2

    jbunniii

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    I'm not sure what you did to get the last line. [itex]a^{ms} = (a^m)^s = e^s = e[/itex], so the last line should be [itex]a = a^{nt}[/itex].
     
  4. Oct 18, 2012 #3

    Zondrina

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    Ohhh whoops. I got confused by all the tags lol.

    So really I have :

    1 = ms + nt
    a1 = ams + nt
    a = ams ant
    a = (am)sant
    a = esant
    a = ant

    Hence a can be written as the nth power of some element in the group and we are done.
     
  5. Oct 18, 2012 #4

    jbunniii

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    Right, specifically [itex]a = (a^t)^n[/itex].
     
  6. Oct 18, 2012 #5

    Zondrina

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    Yeah I wrote that down for my final solution after, I was just highlighting that your hint let me get there nice and quickly.

    I wish I had analysis homework rather than this. I feel lost with algebra all the time.
     
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