My friend after some time in industry is back in school, currently taking graduate algebra. I was today looking at one of his homework and in particular, I thought about and worked out one of the problems, which is to prove the uniqueness part of the Jordan-Hölder theorem. Formally, if is a finite group and
are composition series of , then and there exists and isomorphisms .
Suppose WLOG that and as a base case . Then clearly, and if , . must hold as it is normal in . Now, remember there is a theorem which states that if are normal subgroups of with , then . (This follows from , which shows the commutator to be the identity). Thus there are no other normal proper subgroups other than and .
For the inductive step, take . By the second isomorphism theorem, . Take any composition series for to construct another for via . This shows on application of the inductive hypothesis that . One can do the same for . With both our composition series linked to two intermediary ones that differ only between and the common with factors swapped in between those two, our induction proof completes.