= Cauchy theorem for groups
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= Cauchy's theorem for finite groups
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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 tuples
$$
(x_1,\ldots,x_p)\quad\hbox{with}\quad x_1\cdots x_p=1.
$$
The tuple set has size $|G|^{p-1}$, and its fixed tuples are precisely $(x,\ldots,x)$ with $x^p=1$. Counting nonfixed orbits modulo $p$ forces a nonidentity fixed tuple.
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