If a group has one p-sylow subgroups, then this subgroup must be normal.

Question

I've learned that this is true. Why, basically? I'd appreciate you help.

Answer 1

Let $H < G$ be the unique $p$-Sylow subgroup. Let $x \in G$. Then $xHx^{-1}$ is another $p$-Sylow subgroup, so $xHx^{-1} = H$ i.e. $xH=Hx$. This condition for all $x$ implies $H$ is normal.

Answer 2

This is obvious. Any subgroup which has unique order (the only group of such order) is normal.

The intresting in Sylow is the other direction. Namely, if $P$ is a normal subgroup of a group $G$ them it is unique.

This is because all the $p$-Sylow subgroups are conjugated.