# A question to the ascending central chain in modular group algebras

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.