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.

For a prime divisor $p$ of the order of a finite group, rotate the tuples $(g_1,\dots ,g_p)$ whose product is the identity. Rotation preserves that constraint even for a nonabelian group, by conjugating the product. Every nonfixed orbit has $p$ elements, while the fixed tuples correspond to $g^p=1$. There are $|G{|}^{p-1}$ tuples, so the number of fixed tuples is divisible by $p$. Since the identity is one, another solution exists and has order exactly $p$. This supplies an elementary proof of Cauchy theorem for groups.