Exercise 4.5.18 ( A group of order $200$ has a normal Sylow $5$-subgroup)

Proof. Let $G$ be a group of order $|G|=200=2^3\cdot 5^2$ and let $n_5$ be the number of Sylow $5$-subgroups of $G$. By Sylow’s Theorem,

Thus $n_5\in \{1,2,4,8\}$ and $n_5\equiv 1(\mathit{𝑚𝑜𝑑}8)$, thus

By Corollary 20, the unique Sylow $5$-subgroup of $G$ is normal in $G$

A group of order $200$ has a normal Sylow $5$-subgroup. □