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> If $n\_p=1$, then the $p$-Sylow subgroup is normal.
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I've used this fact several times, and it's about time I knew why it's true.
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Let $N$ be a p-sylow subgroup.

For all $x$ ,$xNx^{-1}$ is a conjugate of $N$. If the set of groups conjugate to $N$ has size $1$ we find that $xNx^{-1}=N$ for all $x$.
I don't want to add something different here, but good to know that $n\_p=[G:N\_G(P)]$ when $P$ is a $p$- sylow of $G$. So if $n\_p=1$ then $N\_G(P)=G$ so $P$ is normal in $G$.