Answer to Question #23477 in Abstract Algebra for Tsit Lam

Question

Deduce that, if kH is J-semisimple for any finitely generated subgroup H of G, then kG itself is J-semisimple.

Answer 1

Assume rad kH = 0 for anyfinitely generated subgroup H ⊆ G. For any α∈ rad kG, we have α ∈ kH for somesuch H, so α ∈ kH ∩ rad kG ⊆ rad kH = 0. Thisimplies that rad kG = 0, as desired.