Suppose and are two non-isomorphic simple, nonzero -modules.

Determine all non-trivial submodules of .

Let be a non-trivial submodule of . Note that is a composition series. By Jordan-Holder theorem, all composition series are equivalent and have the same length. Hence must be a composition series too.

Thus or . In particular is simple.

Let and be the canonical projections. Note that is a submodule of , so or . Similarly, or .

By Schur’s Lemma and are either 0 or isomorphisms.

They cannot be both zero since is non-zero. They cannot be both isomorphisms either, as that would imply .

Hence, exactly one of , are zero. So or .