How to Simplify the Term for a Group Exercise?

[Herbststurm]

Hello

I have problems forming a term.

The exercise is:

$$\text{Let G be a group and } x,y,z,u \in G$$

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

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 :)

 

[morphism]

I'm not sure I understand your question. What does "$\mathrm{Z\kern-.3em\raise-0.5ex\hbox{Z}}:$" 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, $(ab)^{-1} = b^{-1} a^{-1}$.

 

[littleHilbert]

lol

I'm not sure I understand your question. What does "" mean?

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.