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# Answer to Question #23885 in Abstract Algebra for john.george.milnor

Question #23885
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 W = V &otimes;k V with the diagonal G-action is not a semisimple kG-module.
1
2013-02-18T10:46:45-0500
With the basis {e1, e2}on V , the G-action is described by (12)e1 = e2,(12)e2 = e1; (123)e1 = e2, (123)e2= e1 + e2. Using these, it is easy to check that the k-spanof x = e1 &otimes; e1, y =e2 &otimes; e2, and z = e1 &otimes; e2 + e2 &otimes; e1 is a 3-dimensional kG-submodule A &sube; W (with B = k &middot; z as a trivial kGsubmodule).We claim that A is not a kG-direct summand of W. Indeed, if W = A &oplus; k &middot; w is a kG-decomposition, we must have w&isin; WG (G-fixed points in W) since G/[G,G]&sim; {&plusmn;1} implies that any 1-dimensional kGmodule is trivial.But if w = a(e1 &otimes; e1) + b(e2 &otimes; e2) + c(e1&otimes; e2) + d(e2 &otimes; e1), (12)w = w implies thatc = d so w &isin; A, a contradiction.This shows that W is notsemisimple. (In fact, (123)w = wimplies further that a = 0, so we have WG = k &middot; z = B.The kG-composition factors of W are the trivial G-modules B,W/A, together with A/B &sim; V .)

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