Corollary

If the order of G is divisible by p, the group G contains an element of order p.

By virtue of Theorem 2, this is reduced to the case where G is a $p$-group $\neq \{e\}$; if $x \in G$ is different from e, the cyclic group generated by x is then of order $p^n$ with $n \geqslant 1$ and it therefore contains a subgroup of order $p$.