# Answer to Question #23478 in Abstract Algebra for Tsit Lam

Question #23478

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

Expert's answer

Let

*α**∈**rad**kG*. Then*α**∈**kH ∩*rad*kG**⊆**rad**kH*for some finitelygenerated subgroup*H**⊆**G*. Now dim*=*_{k}kH*|H|*is countable, and*k*is uncountable. Therefore,*α*^{n}*= 0 for some**n ≥*1.
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