Answer to Question #23893 in Abstract Algebra for jeremy

Question

Let k be any field of characteristic 3, G = S3 and let V be the kG-module ke1 ⊕ ke2 ⊕ ke3/k(e1 + e2 + e3),
on which G acts by permuting the ei’s. Is G a completely reducible linear group?

Answer 1

Since (123) does not act triviallyon V , the representation homomorphism ϕ : G → GL(V ) must be injective.Therefore, ϕrealizes G as alinear group in GL(V ). It is easy to see that V_o = k(e1− e2) is a kG – module affording the sign representation (of G),with V/V_o affording the trivial representation, so kGV isindecomposable. Therefore, G ⊆GL(V ) is not completelyreducible.