Cauchy's theorem on subgroup existence

https://arbital.com/p/cauchy_theorem_on_subgroup_existence

by Patrick Stevens Jun 18 2016 updated Jun 30 2016

Cauchy's theorem is a useful condition for the existence of cyclic subgroups of finite groups.


Cauchy's theorem states that if is a finite Group and is a prime dividing the order of , then has a subgroup of order . Such a subgroup is necessarily cyclic (proof).

Proof

The proof involves basically a single magic idea: from thin air, we pluck the definition of the following set.

Let the collection of -[-tuple]s of elements of the group such that the group operation applied to the tuple yields the identity. Observe that is not empty, because it contains the tuple .

Now, the cyclic group of order acts on as follows: where is the generator of . So a general element acts on by sending to .

This is indeed a group action (exercise).

%%hidden(Show solution):

Now, fix .

By the Orbit-Stabiliser theorem, the orbit of divides , so (since is prime) it is either or for every .

Now, what is the size of the set ? %%hidden(Show solution): It is .

Indeed, a single -tuple in is specified precisely by its first elements; then the final element is constrained to be . %%

Also, the orbits of acting on partition (proof). Since divides , we must have dividing . Therefore since , there must be at least other orbits of size , because each orbit has size or : if we had fewer than other orbits of size , then there would be at least but strictly fewer than orbits of size , and all the remaining orbits would have to be of size , contradicting that . [todo: picture of class equation]

Hence there is indeed another orbit of size ; say it is the singleton where .

Now acts by cycling round, and we know that doing so does not change , so it must be the case that all the are equal; hence and so by definition of .