I am cramming for my upcoming algebra comprehensive exam so I will probably be making posts like this for a while.
A chain of subgroups of a finite group with simple for all is called a composition series of . (The are called composition factors.) The theorem in question states that every group has a composition series, and the composition factors in any two such series are unique up to reordering.
We can easily prove the first part of the theorem, that any finite group has a composition series, using the extreme principle. Let be the largest integer for which there is a chain of subgroups. (Such an exists because there is trivially a chain of length , and our group is finite.) We claim that the composition factors in this case are simple. If not, WLOG has a normal subgroup . By correspondence, for some . But then is a longer chain, a contradiction.
While the proof of the general theorem requires some non-trivial group theory (e.g. see here), we can easily prove the theorem for finite abelian groups using elementary number theory. For such a group, the orders of the composition factors must exactly be the prime factors of the order of the group—listed with multiplicity. So the result follows immediately from the fundamental theorem of arithmetic.