I refer to this site: http://planetmath.org/groupsoforderpq
In case 2, it is stated that $\text{Aut}(Q)$ has a subgroup $P'$ of order $p$, where $P^{{\prime}}={\{x\mapsto x^{i}\mid i\in\mathbb{Z}/q\mathbb{Z},i^{p}=1\}}$.
May I ask what is the point of showing this? The proof seems to make no further reference to $P'$ after that.
Thanks a lot.