Show that the groups $G$ and $\langle G, * \rangle$ are isomorphic.

Question

Let $G$ be a group and $g\in G$. Define a new operation $*$ on $G$ by $a*b=agb$.

What I have done so far is up to showing that indeed $\langle G,*\rangle$ is a group. When showing that these groups are isomorphic I ended having trouble finding a function.

Any hint will be greatly appreciated.

Answer 1

Consider the map from the original $G$ into $(G,*)$ defined by $h\mapsto g^{-1}h$ and prove that it is a group isomorphism.