# Answer to Question #23881 in Abstract Algebra for Irvin

Question #23881

Let U, V be simple modules of infinite k-dimensions over a group algebra kG (where k is a field), and let W = U ⊗k V , viewed as a kG-module under the diagonal G-action. Does W have finite length?

Expert's answer

The answer is “no”, by acounterexample. Let

G-fixed points in W.

*G*be the multiplicative group of any infinite fieldextension*K*of*k*. Let*U*(resp.*V*) be a copy of*K*(as*k*-vector space) with*G*-action defined by*g**∗**u*=*gu*(resp.*g**∗**v*=*g*^{−}^{1}*v*).Clearly,*U*,*V*are simple*kG*-modules. If we view*K*asa*kG*-module with the trivial*G*-action, the*k*-map*ϕ*:*U**⊗**k V → K*induced by*u**⊗**v → uv*is easily checked to be a*kG*-epimorphism.Since*K*has infinite length as a*kG*-module, so does*W*=*U**⊗**k V*. We finish by checking that*W*is*not*a semisimple*kG*-module. Indeed, if*W*is semisimple,*ϕ*would be a split*kG*-epimormorphism,so*W*would have a submodule with trivial*G*-action mappingisomorphically onto*K*. We have contradiction there are actually nononzeroG-fixed points in W.

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