Cauchy group theorem

- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

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.

- 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)

Last revised on June 30, 2015 at 04:08:40. See the history of this page for a list of all contributions to it.