This is a proof of Cauchy's theorem that does not use Lagrange, it is due to James McKay. It has an uncanny similarity to the proof of Fermat's Little theorem using necklaces.
Let $G$ be a group of order $np$. Then there are $(np)^{p-1}$ solutions to $g_1g_2\dots g_p=1$ since for any values of $g_1,g_2,\dots ,g_{p-1}$ there is a unique inverse for $g_1g_2\dots g_{p-1}$. We call $S$ the set of solutions, we have asserted $|S|$ is a multiple of $p$.
Notice if $g_1,g_2\dots g_p=1$ then $g_ig_{i+1}\dots g_pg_1\dots g_{i-1}=1$ also.
Divide $S$ in rotation classes. Where $s$ is in the same class as $s'$ only if they are rotations of each other. Notice all classes have size $1$ or $p$ (this uses $p$ is prime).Therefore the number of classes of size $1$ is multiple of $p$. Since $\underbrace{1,1\dots ,1}_\text{p times}$ makes up a class of size $1$ there must be another, this provides the desired element of order $p$.
We may now use Cauchy Theorem to determine a group $G$ of order $p$ has an element $g$ of order $p$, this element generates a cyclic subgroup of order $p$. This subgroup must be $G$ so $G$ is cyclic.