Answer to Question #23478 in Abstract Algebra for Tsit Lam
If k is an uncountable field, show that, for any group G, rad kG is a nil ideal.
Let α ∈ rad kG. Then α ∈ kH ∩ rad kG ⊆ rad kH for some finitelygenerated subgroup H ⊆ G. Now dimkkH = |H| is countable, and k is uncountable. Therefore, αn = 0 for some n ≥ 1.
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