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Algebra: abelian problem

  1. Oct 16, 2015 #1
    1. The problem statement, all variables and given/known data
    Let ##x,y,z## be arbitrary integers and the GCD: greatest common divider between any two is ##1##. Let ##(G,\cdot )## be abelian. Prove the implication:
    [tex]
    a,b\in G\left (a^x = b^y = (ab)^z = e\Longrightarrow a=b=e\right )
    [/tex]
    where ##e## is the unit element of ##G##.
    2. Relevant equations
    Corollary: For every integers ##x,y## there exist integers ##u,v## such that ##xu+yv = GCD(x,y)##

    3. The attempt at a solution
    If we look at what ##b## is equal to:
    [tex]
    b^y = e \Leftrightarrow b^yb^{1-y} = eb^{1-y} \Leftrightarrow b^1 = b^{1-y}
    [/tex]
    As ##x,y,z## have no common dividers, they have to be nonzero otherwise, for instance ##GCD(x,0) = x##, which is not necessarely ##1##.
    At this point I'm stuck. If ##b^1 = b^{1-y}## and provided also ##y\neq 0##, then ##b\in G## is equal to one of its powers. Cyclicity?
     
  2. jcsd
  3. Oct 16, 2015 #2

    RUber

    User Avatar
    Homework Helper

    Have you looked at ## e^{xy}=e^{yz} = e^{xz} = a^{xyz} = b^{xyz} = (ab)^{xyz} ##?
     
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