Many texts prove the theorem with the use of strong induction and the class equation, though considerably less machinery is required to prove the theorem in the abelian case. One can also invoke group actions for the proof.
Proof 1
We first prove the special case that where is abelian, and then the general case; both proofs are by induction on = ||, and have as starting case = which is trivial because any non-identity element now has order. Suppose first that is abelian. Take any non-identity element, and let be the cyclic group it generates. If divides ||, then ||/ is an element of order. If does not divide ||, then it divides the order of the quotient group /, which therefore contains an element of order by the inductive hypothesis. That element is a class for some in, and if is the order of in, then = in gives = in /, so divides ; as before / is now an element of order in, completing the proof for the abelian case. In the general case, let be the center of, which is an abelian subgroup. If divides ||, then contains an element of order by the case of abelian groups, and this element works for as well. So we may assume that does not divide the order of ; since it does divide ||, there is at least one conjugacy class of a non-central element whose size is not divisible by. But the class equation shows that size is , so divides the order of the centralizer of in, which is a proper subgroup because is not central. This subgroup contains an element of order by the inductive hypothesis, and we are done.
Proof 2
This proof uses the fact that for any action of a group of prime order, the only possible orbit sizes are 1 and, which is immediate from the orbit stabilizer theorem. The set that our cyclic group shall act on is the set of -tuples of elements of whose product gives the identity. Such a -tuple is uniquely determined by all its components except the last one, as the last element must be the inverse of the product of those preceding elements. One also sees that those elements can be chosen freely, so has ||−1 elements, which is divisible by. Now from the fact that in a group if = then also =, it follows that any cyclic permutation of the components of an element of again gives an element of. Therefore one can define an action of the cyclic group of order on by cyclic permutations of components, in other words in which a chosen generator of sends. As remarked, orbits in under this action either have size 1 or size. The former happens precisely for those tuples for which =. Counting the elements of by orbits, and reducing modulo, one sees that the number of elements satisfying = is divisible by. But = is one such element, so there must be at least other solutions for, and these solutions are elements of order. This completes the proof.
Uses
A practically immediate consequence of Cauchy's theorem is a useful characterization of finite -groups, where is a prime. In particular, a finite group is a -group if and only if has order for some natural number. One may use the abelian case of Cauchy's Theorem in an inductive proof of the first of Sylow's theorems, similar to the first proof above, although there are also proofs that avoid doing this special case separately.
