# centralizer of transvection

let $T=T_{a,u}(v)$ be transvection, I want to find centralizer of $T_{a,u}(v)$ in $GL(V)$. What is $C _{GL(V)}(T)$?

$T_{a,u}(v)=v+u(v)a$ where $a$ is vector in vector space $V$ on field $F$ and $u$ is linear functional on $V$.It is not hard to see $T_{a,u}(v)\in SL(V)$

I have not any idea how to deal with it.

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The centralizer in what group? – Arturo Magidin Mar 26 '12 at 19:06

You can choose a basis of $V$ such that the matrix of the transvection is the identity matrix with an extra 1 in the $(1,2)$-position. It centralizer consists of all non-singular matrices $(a_{ij})$ with $a_{11}=a_{22}$, $a_{i1}=0$ for all $i \ne 1$ and $a_{2j}=0$ for all $j \ne 2$.