# 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 &otimes;k V , viewed as a kG-module under the diagonal G-action. Does W have finite length?
1
2013-02-19T10:43:49-0500
The answer is &ldquo;no&rdquo;, by acounterexample. Let 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 &lowast; u = gu (resp. g &lowast; v = g&minus;1v).Clearly, U, V are simple kG-modules. If we view K asa kG-module with the trivial G-action, the k-map
ϕ : U &otimes;k V &rarr; K induced by u &otimes; v &rarr; uv is easily checked to be a kG-epimorphism.Since K has infinite length as a kG-module, so does W = U&otimes;k V . We finish by checking that W is nota 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 nononzero
G-fixed points in W.

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