I working on a problem I can do since I have done many of this forum. However, I don't understand the group mapping:
G is a group and we are mapping G to G by $\displaystyle g\to g^{-1}$.
How does this mapping work?
I don't understand the exact meaning of your question. The mapping $\displaystyle \psi:G\to G$ , $\displaystyle \psi(g)=g^{-1}$ is well defined (for every $\displaystyle g\in G$ there exists its inverse $\displaystyle g^{-1}$ and it is unique). Besides, $\displaystyle \psi(ab)=(ab)^{-1}=b^{-1}a^{-1}=\psi(b)\psi(a)$ for all $\displaystyle a,b\in G$ .