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 ).

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*).

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