Corollary of Lagrange's theorem

Order of $g$. Let $d$ be the order of $g$ (the smallest positive integer with $g^d=e$). Then $\langle g\rangle=\{e,g,\ldots,g^{d-1}\}$ has order $d$.

Lagrange. By Lagrange's theorem, $d=|H|$ divides $|G|=n$.

Conclusion. $n=kd$ for some $k$, so $g^n=g^{kd}=(g^d)^k=e^k=e$.

Corollary. In $(\mathbb{Z}/p\mathbb{Z})^*$ with $p$ prime, for any $a\not\equiv0$, $a^{p-1}\equiv1\pmod p$ (Fermat's little theorem).