Groups of prime order

Order of $\langle g\rangle$. By Lagrange, $|\langle g\rangle|$ divides $p$. Since $g\neq e$, $|\langle g\rangle|>1$, so $|\langle g\rangle|=p$.

Conclusion. $\langle g\rangle=G$, so $G$ is cyclic.

Remark. Every group of prime order is isomorphic to $(\mathbb{Z}/p\mathbb{Z},+)$.