Thanks for any help or comments.

In my research I need a characterization for special linear groups over finite fields by some information of its subgroups, especially centralizers. I saw the following on page 56 of the book The Classification of the Finite Simple Groups by Gorenstein, Lyons and Solomon.

Let $G$ be a finite group such that for $x,y\in G$ and $K,J$ are components of $C_G(x), C_G(y)$ respectively. If we have the following properties, then $G$ is isomorphic to $SL_n(F)$.

1) $K,J\cong SL_{n-1}(F)$.

2) $I:=E(K\cap J)\cong SL_{n-2}(F)$.

3) $C_G(K), C_G(J)$ have cyclic Sylow $p$-subgroups.

4) $I$ is contained in a single component of $C_G(u)$ for every $u\in \langle x,y\rangle$

Do you think the above argument is true or needs some new properties or there exists a better characterization?

neighbourhoodof $(x,K)$. This last concept is something I'm not familiar with... $\endgroup$ – Nick Gill Sep 24 '15 at 9:03