nLab
Cauchy group theorem

Statement

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

Theorem

(Cauchy)

If a prime number $p$ divides $\mid G\mid$, then equivalently

• $G$ has an element of order $p$;

• $G$ has a subgroup of order of a group $p$.

References

• James McKay, Another proof of Cauchy’s group theorem, American Math. Monthly, 66 (1959), p. 119.