57 506
Assignments Done
Successfully Done
In February 2018
Your physics homework can be a real challenge, and the due date can be really close — feel free to use our assistance and get the desired result.
Be sure that math assignments completed by our experts will be error-free and done according to your instructions specified in the submitted order form.
Our experts will gladly share their knowledge and help you with programming homework. Keep up with the world’s newest programming trends.

Answer on Abstract Algebra Question for john.george.milnor

Question #23886
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.
Expert's answer
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, dimk 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.

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!


No comments. Be first!

Leave a comment

Ask Your question