Group order

Cauchy’s order theorem

Let be a finite group and be a prime dividing . Then has an element of order .^[MATH4031] group

Proof via permutation groups (James McKay)

Let

Note is closed under the natural action of , since if , then .

By the Orbit-stabilizer theorem, a -orbit in has size 1 or . For an element to have an orbit of size 1, it must have order 1 or .

Furthermore, , by basic combinatorics (the first choices are free).

It follows that the number of orbits of size 1 is divisible by , and hence there exists more than 1 orbit of size 1. Since only one of these may be the repeated identity, it follows there exists at least one element of order .

---

tidy | en | sembr