# Answer to Question #23475 in Abstract Algebra for Tsit Lam

Question #23475

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.

Expert's answer

Take a

so

*kG*-epimorphism*ϕ*:*F → V*, where*F*is asuitable free*kG*module, and let*E*= ker(*ϕ*). Since*V*is projective as a*k*-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.Need a fast expert's response?

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