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Question #18097

Show that if k0 is any finite field and G is any finite group, then (k0G/rad k0G) ⊗k0 K is semisimple for any field extension K ⊇ k0.

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**1.**Let H be a normal subgroup of G. If H is finite, show that I is also nilpotent.**2.**Let H be a normal subgroup of G. If rad kH is nilpotent, show that I is also nilpotent.**3.**Let H be a normal subgroup of G. Show that I = kG • rad kH is an ideal of kG.**4.**If k is an uncountable field, show that, for any group G, rad kG is a nil ideal.**5.**For any subgroup H of a group G, show that if kH is J-semisimple for any finitely generated subgroup**6.**For any subgroup H of a group G, show that kH ∩ U(kG) ⊆ U(kH) and kH ∩ rad**7.**Let G be a finite group whose order is a unit in a ring k, and let W ⊆ V be left kG-modules

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