3
$\begingroup$

Let $p$ be a prime number, $G=:G_1$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_2:=1+\operatorname{rad}(KG)$ is a p-group containing $G$.

My questions are:

1.) When does it happen that the derived length of $G_2$ is equal to the derived length of $G_1$?

2.) When does it happen that the Lie derived length of $KG_2$ is equal to the Lie derived length of $KG_1$?

Conjecture: If the length are equal, then the size of the derived subgroup of $G$ is bounded.

Improve this question
$\endgroup$
5
  • 1
    $\begingroup$ If a group has cyclic center of order $p^k$ and is of class 2, then derived length of group ring is $\rm{log}_2(p^k +1)$ rounded up. Maybe that could give easy examples for second question. $\endgroup$
    – Denis T
    Commented Sep 27, 2019 at 21:38
  • $\begingroup$ Thanks I know this theorem. A similar one is also valid for the group of units. $\endgroup$
    – Sven Wirsing
    Commented Sep 28, 2019 at 8:59
  • $\begingroup$ What do you mean by "on what terms"? Do you mean "When does it happen that"? $\endgroup$
    – YCor
    Commented Oct 3, 2019 at 14:25
  • $\begingroup$ Yes, I have adjusted the questions accordingly. $\endgroup$
    – Sven Wirsing
    Commented Oct 3, 2019 at 17:46
  • $\begingroup$ we should close the topic. $\endgroup$
    – Sven Wirsing
    Commented Dec 16, 2020 at 20:05

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .