# Answer to Question #23892 in Abstract Algebra for jeremy

Question #23892

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 ).

on which G acts by permuting the ei’s. Show that this realizes G as a linear group in GL(V ).

Expert's answer

Since (123) does not act triviallyon

*V*, the representation homomorphism*ϕ*:*G →*GL(*V*) must be injective.Therefore,*ϕ*realizes*G*as alinear group in GL(*V*).Need a fast expert's response?

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