Let $G$ be a simple group of lie type over a field of odd characteristic $p$, and $u$ is a regular unipotent element of $G$, that is a $p$-element with centralizer of minimum rank.

We know that centralizer of a regular unipotent element has to be a $p$-group.

We also know that every two regular unipotent elements of $G$ are conjugate, so there is only one conjugacy class of regular unipotent elements in $G$.

But what else we can say? Is it true that size of centralizer of a regular unipotent element divides size of centralizer of other unipotent elements?


Actually it is not my answer! Some one answered this question to me. In $PSL(4,3)$ the order of $C_G(x)$ is $\{ 5832,~1944,~81, ~72, ~36,~12 \}$ where $x$ is a $p$-element. It is obvious that there is no element which its centralizer divides the others.


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