Cauchy theorem for groups

If a prime divides the order of a finite group, the group contains an element of that prime order.

One proof lets a cyclic group of order $p$ rotate the tuplesThe tuple set has size $|G{|}^{p-1}$, and its fixed tuples are precisely $(x,\dots ,x)$ with $x^p=1$. Counting nonfixed orbits modulo $p$ forces a nonidentity fixed tuple.