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!
Best Answer
The argument shows that for any $p$-Sylow subgroup $H$, there exists $Q\in S$ such that $H=Q$. But $S$ was by definition the set of conjugates of $P$, so this means $H$ is conjugate to $P$. So, this proves that every $p$-Sylow subgroup $H$ is conjugate to $P$, which proves (ii).
Moreover, the argument shows that if $H$ is $p$-Sylow and $Q$ is any fixed point of the action of $H$ on $S$, then $H=Q$. So, there can only be one such fixed point, namely $H$.