Originally published at 狗和留美者不得入内. You can comment here or there.
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
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.
