The Frattini subgroup of a finite group $G$ is the intersection of all its proper maximal subgroups. Let $p$ be a prime number. Show that the Frattini subgroup of $\mathbb{Z} / p^{n}$, $n \geq 2$, is generated by $p$.
The Frattini subgroup of a finite group $G$ is the intersection of all its proper maximal subgroups. Let $p$ be a prime number. Show that the Frattini subgroup of $\mathbb{Z} / p^{n}$, $n \geq 2$, is generated by $p$.