Conjugacy Classes and Group Order

[Kalinka35]

Homework Statement

Is there a proof that the number of elements in a conjugacy class of a group has to divide the order of the group?
Everyone seems to cite it left and right but I've not seen a proof of it anywhere.

Homework Equations

The Attempt at a Solution

I'm guessing Langrange's Theorem may come into play but since conjugacy classes themselves aren't subgroups I'm not sure how.

 

[matt grime]

It is a trivial* consequence of the orbit-stabilizer theorem, or the class equation if you want to be fancy.

If G is a finite group acting on a set X, then |G|=|Orb(x)||Stab(x)|. Let X be G, and let the action be conjugation.

* Trivial in the sense of an immediate and obvious corollary, not something that is unimportant and easy.