Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $1+\operatorname{rad}(KG)$ is a p-group containing $G$. $G$ is normal in $1+\operatorname{rad}(KG)$ if and only if $G$ is abelian. As $1+\operatorname{rad}(KG)$ is nilpotent, the group $G$ is subnormal in $1+\operatorname{rad}(KG)$. Is the defect of subnormality known for $G$ in $1+\operatorname{rad}(KG)$?

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    $\begingroup$ adjusted with \operatorname{rad} $\endgroup$ – Sven Wirsing Apr 16 at 19:16

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