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Group - How to transform?

  1. Nov 6, 2008 #1
    Hello

    I have problems forming a term.

    The exercise is:

    [tex] \text{Let G be a group and } x,y,z,u \in G [/tex]

    [tex]\mathrm{Z\kern-.3em\raise-0.5ex\hbox{Z}}: ~ \left(x \left( \left( \left( y^{-1} \left( x^{-1} \cdot z \right) \right) ^{-1} \cdot u \right) \cdot \left( y \cdot u \right)^{-1} \right) ^{-1} \right) = z [/tex]

    I kwon that I have to show that:

    i.) Associative Law

    ii.) Identity Element

    iii.) Inverse Element

    If I look the term it is clear that I have to form it such that I only have z=z and the other elements x,y,u should be transformed into the identity because of their inverse elements.

    I don't know how to form this concretely

    Thanks for help
    Greetings

    p.s.
    This is not homework or something like that. I want to dish my mathematical tools :)
     
  2. jcsd
  3. Nov 7, 2008 #2

    morphism

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    Science Advisor
    Homework Helper

    I'm not sure I understand your question. What does "[itex]\mathrm{Z\kern-.3em\raise-0.5ex\hbox{Z}}:[/itex]" mean?

    I'm going to assume you want to show that the equation you posted holds for all x,y,z,u. In which case, notice that, for any a,b in G, [itex](ab)^{-1} = b^{-1} a^{-1}[/itex].
     
  4. Nov 7, 2008 #3
    lol
    This is German and means "zu zeigen" - "has to be shown". This abbr. is often in the beginning of a theorem or an implication in it.
     
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