Let C be the complex numbers and let G be a group so CG is a group algebra.
Let V, W be CG-modules and let f:V->W be a module homomorphism. Prove that there exists a submodule U of V such that V is the drect sum of U and ker(f), and U is isomorphic to im(f).
My thoughts: If we knew V was simple, then since ker(f) is a submodule, we would know it has a complement U. So 2 issues: Is V simple?, and how do we know U is isomorphic to im(f)?
Thanks
Let V, W be CG-modules and let f:V->W be a module homomorphism. Prove that there exists a submodule U of V such that V is the drect sum of U and ker(f), and U is isomorphic to im(f).
My thoughts: If we knew V was simple, then since ker(f) is a submodule, we would know it has a complement U. So 2 issues: Is V simple?, and how do we know U is isomorphic to im(f)?
Thanks