Example

(1) In the group $\mathfrak{S}_p$ let $\zeta$ be a cycle of order $p$. The subgroup generated by $\zeta$ is a Sylow $p$-subgroup of $\mathfrak{S}_p$ for $p$ does not divide $(p - 1)!$. (2) Let k be a finite field of characteristic $p$ and let n be a positive integer. The strict triangular group $T_1(n, k)$ is a Sylow $p$-subgroup of the group $\mathbf{GL}(n, k)$.