One approach to proving results about finite groups is by using induction on the order of the group. The general idea is that one has many constructions of subgroups, such as the formation of the centre, or of centralizers of elements, or of normalizers of subgroups, to which one can hope to apply an inductive argument.
Basic group theory.
Cauchy's theorem states that if a prime divides the order of a finite group , then contains an element of order . We present a proof of Cauchy's theorem via induction on the order of . (Many other proofs are possible; a simple proof using an approach via group actions is here.)
We first consider the decomposition of into conjugacy classes. If is any element of , then the stablizer of under the action of via conjugation is equal to , the centralizer of in . Thus the number of conjugates of is equal to the index and so if is a set of conjugacy class representatives of , then
Note that if and only if , or equivalently, if and only if lies in the centre of . Thus if we label our conjugacy class representatives so that the first representatives are the elements of the centre, then we may rewrite the above equation in the form
where the sum is over the non-central conjugacy class representatives, i.e. over those conjugacy class representatives whose conjugacy class contains more than one element. (This formula is known as the class equation.)
Now suppose that does not divide for some . Then since does divide , we find that divides the order . Since is not central in (by assumption), we find that is a proper subgroup of . Thus, by induction, we may conclude that contains an element of order . Since is a subgroup of , we also get an element of order in , and so are done.
It remains to consider the case when divides for all . Since divides the order of , we then conclude from the class equation that divides . Since is an abelian group, this reduces Cauchy's theorem for a general finite group to the case of a finite abelian group.
Cauchy's theorem for finite abelian groups follows immediately from the fundamental theorem of finite abelian groups. However, we can also prove it directly, using the same strategy of induction on the order.
Thus suppose that is a finite abelian group, and that is a prime dividing . Let be any non-identity element of , and let be the subgroup of generated by . The order of is equal to the order of , which we denote by . If divides , then is an element of , and hence of , of order , and we are done.
If does not divide the order of , then we form the quotient . (It is here that we use the fact that is abelian, so as to be certain that its subgroup is normal.) Since was chosen to be non-trivial, the order of is less than the order of , and since does not divide the order of by assumption, it must divide the order of . By induction, we conclude that must have an element of order , say . Let be a preimage of under the natural projection (or, if you prefer, a representative of the coset ). Since the image of in has order , the order of must be divisible by . Thus, if we let denote this order, the element of has order . This completes the proof of Cauchy's theorem in the abelian case.