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. Show that this realizes G as a linear group in GL(V ).

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).