Question

If k is an uncountable field, show that, for any group G, rad kG is a nil ideal.

Answer 1

Let α ∈ rad kG. Then α ∈ kH ∩ rad kG ⊆ rad kH for some finitelygenerated subgroup H ⊆ G. Now dim_kkH = |H| is countable, and k is uncountable. Therefore, αⁿ = 0 for some n ≥ 1.