Consider $G_n=\{ \overline{x}\in\mathbb{Z}/n\mathbb{Z}; \gcd(x,n)=1 \}$, and $q$ an odd prime number.
For all $x\in \mathbb{Z}$, $\overline{x}$ represents, in this question, the class of $x$ modulo $q^\alpha$, and let's use the notation $\dot{x}$ for the class of $x$ modulo $q$ and consider the morphism of groups \begin{align*} \Psi : G_{q^\alpha} &\rightarrow G_q \\ \overline{x}&\mapsto \dot{x}. \end{align*} Consider $\overline{x}\in G_{q^\alpha}$ such as $\dot{x}$ is a generator of $G_q$. Denote for $\langle\overline{x}\rangle$ the sub group of $G_{q^\alpha}$ generated by $\overline{x}$. Show that exists one element $\overline{y}\in \langle\overline{x}\rangle$ of order $q-1$.
It's a long question, having trouble in this part.
$\dot{x}$ is a generator of $G_q$ is equivalent to order of $\dot{x}$ is $\varphi(q) = q - 1$. So order of $\bar{x}$ in $G_{q^{\alpha}}$ is divided by $q-1$ and divide order of $G_{q^{\alpha}}$ which is $\varphi(q^{\alpha}) = (q-1)q^{\alpha - 1}$. We deduce that order of $\bar{x}$ is of form $q^{\beta}(q-1),\, 0\leq \beta \leq\alpha - 1$. Now take $\bar{y} = \bar{x}^{q^{\beta}}$