Corollary 1

Let P be a Sylow p-subgroup of G, let N be its normalizer in G and let M be a subgroup of G containing N. The normalizer of M in G is equal to M.

Let $s \in G$ be such that $sMs^{-1} = M$. The subgroup $sPs^{-1}$ of M is a Sylow $p$-subgroup of M. There thus exists $t \in M$ such that $sPs^{-1} = tPt^{-1}$; then $t^{-1}s \in N$, whence $s \in tN \subset M$.