Let $G$ be a finite group with order ${\vert G\vert} \in \mathbb{N}$.
This result is not completely trivial. One route to this would go as follows: knowing that Sylow $p$-subgroups $H$ of $G$ exist (see class equation for a proof), any nontrivial element $h$ of $H$ would be of order $p^r$ for some $r \gt 0$, and then $h^{p^{r-1}}$ would be the desired element. Come to think of it, it’s actually an immediate consequence of the theorem here. But see McKay for a snappier proof.
Cauchy had claimed a proof of his eponymous theorem in 1845, but in fact his proof had a gap. See Meo for a historical discussion.
Name after Augustin Cauchy.
James McKay, Another proof of Cauchy’s group theorem, American Math. Monthly, 66 (1959), p. 119.
M. Meo, The mathematical life of Cauchy’s group theorem, Historia Mathematica Volume 31, Issue 2 (May 2004), 196–221. (web)
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