> 
> If $n\_p=1$, then the $p$-Sylow subgroup is normal.
> 
> 
> 

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 $x$.
If I may add something.
The following statement is true:

Let $U\leq G$ be a subgroup such that there is no other subgroup that has the same order. Then $U$ is a characteristic subgroup. That is, *every automorphism* of $G$ maps $U$ to itself.