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 .