nLab Cauchy group theorem

Statement

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

Theorem

(Cauchy)

If a prime number $p$ divides ${\vert G\vert}$, then equivalently

• $G$ has an element of order $p$;

• $G$ has a subgroup of order $p$.

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.

Remark

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.

Related concepts

• Cauchy's theorems

References

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)