nLab Cauchy group theorem




Let GG be a finite group with order |G|{\vert G\vert} \in \mathbb{N}.



If a prime number pp divides |G|{\vert G\vert}, then equivalently

This result is not completely trivial. One route to this would go as follows: knowing that Sylow pp-subgroups HH of GG exist (see class equation for a proof), any nontrivial element hh of HH would be of order p rp^r for some r>0r \gt 0, and then h p r1h^{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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