(From Rotman, "Introduction to the Theory of Groups") I am not really sure where to start with this one.
I know that characteristically simple groups can be expressed as internal direct products of pairwise isomorphic, finite simple groups. And that we can find a basis for any subspace, $U$ of $V$ that can be extended to a subspace $W$ for the whole space so that $V=U\oplus W$.
But I'm not sure how to translate between the language of groups and the language of vector spaces in this proof.
Surely we can't just use $(U,+)$ and $(W,+)$ as direct factors, as they may not be isomorphic!