I am cramming for my algebra comprehensive exam so I will probably be posting stuff 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 with , and our group is finite.) We claim that the composition factors in this case are simple. If not, WLOG has a non-trivial 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—listed with multiplicity—of the order of the group. So the result follows immediately from the fundamental theorem of arithmetic.