Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $char(K)=p$. It is well-known that the group $1+rad(KG)$ is a p-group containing $G$. Let us focus on the sequence $G\cdot Z_n(1+rad(KG))$ for $n\in \mathbb{N}$ where $Z_n(1+rad(KG))$ is the n-th memeber of the ascending centrali chain of $1+rad(KG)$.

My question is what the minimal $n$ is such that $G\cdot Z_n(1+rad(KG))$ reaches $1+rad(KG)$. Of course, this minimal $n$ is not greater than the class of nilpotency of $1+rad(KG)$.

My conjecture is that is identical to the class of nilpotency.


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