Answer to Question #23483 in Abstract Algebra for sanches

Abstract Algebra

Question #23483

Let k ⊆ K be two fields and G be a finite group. Show that rad (KG) = (rad kG) ⊗k K.

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Related Questions

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