Answer to Question #23886 in Abstract Algebra for john.george.milnor

Question

For G = S3 and any field k of characteristic 2, view V = ke1 ⊕ ke2 ⊕ ke3/k(e1 + e2 + e3) as a (simple) kG-module with the permutation action. Show that kG ∼ M2(k)×(k[t]/(t2)), and that kG/rad(kG) ∼ M2(k)×k.

Answer 1

Let ϕ : kG → End_k(V) ∼ M₂(k) be the k-algebrahomomorphism associated with the kG-module V . This map is ontosince V is absolutely irreducible. Thus, dim_k ker(ϕ) = 2 .Nowe = (1) + (123) +(132) is a central idempotent of kG, with kG · e = ke ⊕ kσ, where σ =(sum over g∈G)g ∈ kG. Since σ₂ = 6σ =0, we have kG · e ∼ k[t]/(t²). By a simple computation, ϕ(e) = 0, so kG · e =ker(ϕ) (both spaces being 2-dimensional).Therefore, kG = kG · (1 − e) × kG · e ∼ M2(k) × (k[t]/(t2)).Computing radicals, we get rad(kG) = rad(kG · e) = k · σ, so kG/rad(kG) ∼ M₂(k) × k.