Let be p a prime number and let be $|x|=p^{n-1}$.
I consider $\alpha \in {\rm Aut}(\langle x \rangle)$ defined in this way: $\alpha : x \mapsto x^{1+p}$.
Let be $\gamma=\alpha^{p^{n-3}}$.
I don't understand why $\gamma: x \mapsto x^{1+p^{n-2}}$.
Why is it not $\gamma(x)=x^{{(1+p)}^{{p}^{n-3}}}$?
Thanks for the help!