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?

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    $\begingroup$ The nice thing about involutions is that any two of them generate a dihedral subgroup. If its order is $2n$ with $n$ odd, then both involutions are conjugated to each other (in the dihedral subgroup, hence also in $G$). If $n$ is even, the center of the dihedral subgroup is $Z_2$, therefore it contains an involution whose centralizer contains both involutions. In both cases you got some nontrivial information for free. (For an application for infinite groups see my answer to Is there a group which has precisely all finite groups as subgroups?.) $\endgroup$ – j.p. Nov 2 '15 at 14:52

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