= Cyclic-tuple proof of Cauchy theorem
{title2=$\#\{g:g^p=1\}\equiv |G|^{p-1}\pmod p$}
For a prime divisor $p$ of the order of a <finite group>, rotate the tuples $(g_1,\ldots,g_p)$ 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>.
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