This is the excerpt of the proof of Sylow's theorem from Lang's and some moments of the are unclear to me.
1) In order to prove (ii) they take $H$ to be Sylow $p$-subgroup and it is not obvious to me why any two Sylow $p$-subgroups are conjugate?
2) Is $H$ is a Sylow $p$-subgroup then why it has only one fixed point?
I have spent some hours in order to understand these questions but was not able.
Would be very grateful for detailed help!