1. Consider the set $G=(1, \infty)$ and the function $f: G \rightarrow \mathbb{R}_{+}^{*}, f(x)=x-1$.

Determine a composition law ,,*' defined on $G$ such that $(G, *)$ is a group, and the function $f$ is an isomorphism from the group $(G, *)$ to the group $\left(\mathbb{R}_{+}^{*}, \cdot\right)$.
1. $f$ bijective, hence invertible. From $f(x * y)=f(x) f(y)=x y-x-y+1$ it follows that $x * y=f^{-1}(x y-x-y+1)$.

From $f(x)=x-1$ it follows that $f^{-1}(x)=x+1$ and $x * y=x y-x-y+2, x, y \in G$. $2 \text{p}$

It is verified that $(G, *)$ is a group.