I read that for the classification of the finite simple groups the centralizers of involutions plays a crucial role. This has to to with the Brauer-Fowler-Theorem, which in one formulation could be read in this paper by R. Solomon:
Let $G$ be a finite simple group of even order containing an involution $t$. If $|C_G(t)| = c$, then $|G| \le (c!)^2$.
So I am asking could anyone give a lightweight introduction in what sense these centralizers "control the structure of the group", maybe point to simple properties, or even give some accessable, preferable short proofs, about these centralizers or where they were used? So someone not having read all these heavy papers can gain some intuition about them and why they are important?