Answer on Abstract Algebra Question for john.george.milnor
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.
Let ϕ : kG → Endk(V) ∼ M2(k) be the k-algebrahomomorphism associated with the kG-module V . This map is ontosince V is absolutely irreducible. Thus, dimkker(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.
Thanks for the helpp you guys have been big boost for me not just for solution, but for understading the assigment with details step its been very helpfull for me thanks a lot.
Very high quality service everytime.
It will not be my last time. and again i cant thanks you guys enough.