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

Question #23886
For G = S3 and any field k of characteristic 2, view V = ke1 &oplus; ke2 &oplus; ke3/k(e1 + e2 + e3) as a (simple) kG-module with the permutation action. Show that kG &sim; M2(k)&times;(k[t]/(t2)), and that kG/rad(kG) &sim; M2(k)&times;k.
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2013-02-14T08:02:44-0500
Let ϕ : kG &rarr; Endk(V) &sim; 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 &middot; e = ke &oplus; k&sigma;, where &sigma; =(sum over g&isin;G)g &isin; kG. Since &sigma;2 = 6&sigma; =0, we have kG &middot; e &sim; k[t]/(t2). By a simple computation, ϕ(e) = 0, so kG &middot; e =ker(ϕ) (both spaces being 2-dimensional).Therefore, kG = kG &middot; (1 &minus; e) &times; kG &middot; e &sim; M2(k) &times; (k[t]/(t2)).Computing radicals, we get rad(kG) = rad(kG &middot; e) = k &middot; &sigma;, so kG/rad(kG) &sim; M2(k) &times; k.

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