Answer to Question #23475 in Abstract Algebra for Tsit Lam

Question

Let G be a finite group whose order is a unit in a ring k, and let W ⊆ V be left kG-modules. If V is projective as a k-module, then V is projective as a kG-module.

Answer 1

Take a kG-epimorphism ϕ : F → V , where F is asuitable free kGmodule, and let E = ker(ϕ). Since V is projective as ak-module, E is a direct summand of F as k-modules.Then E is a direct summand of F as kG-modules. Thus, V isisomorphic to a direct kG-complement of E in F,
so V is a projective kG-module.