# Answer to Question #23481 in Abstract Algebra for sanches

Question #23481

Let H be a normal subgroup of G. If H is finite, show that I is also nilpotent.

Expert's answer

Any

*σ**∈**G*defines a conjugation automorphism on thesubring*kH**⊆**kG*, and this automorphism must takerad*kH*to rad*kH*. Therefore, (rad*kH*)*σ**⊆**σ**·*rad*kH**⊆**I,*which shows that*I*is anideal of*kG*. This method also shows that*I*=^{n}*kG ·*(rad*kH*)*for any*^{n }*n ≥*1, so if*H*is finite,then*I*is nilpotent..
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