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Question #18099

Let k ⊆ K be two fields and G be a finite group. Show that a kG-module M is semisimple iff the KG-module MK = M ⊗k K is semisimple.

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**1.**Let k ⊆ K be two fields and G be a finite group. Show that rad (KG) = (rad kG) ⊗**2.**Show that if k0 is any finite field and G is any finite group, then (k0G/rad k0G) ⊗k0 K i**3.**Let H be a normal subgroup of G. If H is finite, show that I is also nilpotent.**4.**Let H be a normal subgroup of G. If rad kH is nilpotent, show that I is also nilpotent.**5.**Let H be a normal subgroup of G. Show that I = kG • rad kH is an ideal of kG.**6.**If k is an uncountable field, show that, for any group G, rad kG is a nil ideal.**7.**For any subgroup H of a group G, show that if kH is J-semisimple for any finitely generated subgroup

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