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.