Disclaimer: my question is different from this question.
I am following these math notes on abstract algebra that I found online to teach myself. On page 21 the group action of conjugation is defined as: $$\mu:(x,y)\mapsto x^{-1}*y*x$$ In the other question it is defined as: $$\mu_\text{(other question)}:(x,y)\mapsto x*y*x^{-1}$$ Which works out.
Since we require: $$\mu(x*y,z)=\mu(x,\mu(y,z))$$ I find that the LHS is: $$\mu(x*y,z)=(x*y)^{-1}*z*(x*y)=y^{-1}*x^{-1}*z*x*y$$ Whereas the RHS is: $$\mu(x,\mu(y,z))=\mu(x,y^{-1}*z*y)=x^{-1}*y^{-1}*z*y*x$$ So clearly $$\mu(x*y,z)\neq\mu(x,\mu(y,z))$$
Is there a mistake on my end or did the author mean to define it as: $\mu:(x,y)\mapsto x*y*x^{-1}$?