In this MO thread https://mathoverflow.net/questions/38161/heuristic-argument-that-finite-simple-groups-ought-to-be-classifiable, Borcherds says
One problem, as least with the current methods of classification via centralizers of involutions, is that every simple group has to be tested to see if it leads to new simple groups containing it in the centralizer of an involution. For example, when the baby monster was discovered, it had a double cover, which was a potential centralizer of an involution in a larger simple group, which turned out to be the monster.
The basic theorem about centralizers and involutions is https://en.wikipedia.org/wiki/Brauer%E2%80%93Fowler_theorem, whose Wikipedia page states
Perhaps more important is another result that the authors derive from the same count of involutions, namely that up to isomorphism there are only a finite number of finite simple groups with a given centralizer of an involution. This suggested that finite simple groups could be classified by studying their centralizers of involutions, and it led to the discovery of several sporadic groups. Later it motivated a part of the classification of finite simple groups.
Here are some neat write-ups on Brauer-Fowler that I thought were nice: http://thomasbloom.org/notes/brauer.html, https://terrytao.wordpress.com/2013/05/02/quasirandom-groups-and-a-cheap-version-of-the-brauer-fowler-theorem/.
Looking at https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups#Timeline_of_the_proof, we see that
1955 The Brauer–Fowler theorem implies that the number of finite simple groups with given centralizer of involution is finite, suggesting an attack on the classification using centralizers of involutions.
...
1957 Suzuki shows that all finite simple CA groups of odd order are cyclic.
...
1960 Feit, Marshall Hall, and Thompson show that all finite simple CN groups of odd order are cyclic.
...
1963 Feit and Thompson prove the odd order theorem.
Where CA and CN stand for "all centralizers are abelian" and "all centralizers are nilpotent". This article https://www.mat.uniroma2.it/~eal/Suzukithm.pdf points out the relation between the CA, CN, and Feit-Thompson theorems:
- The CA theorem tells us that groups whose non-identity elements have abelian centralizers are solvable.
- The CN theorem tells us that groups whose non-identity elements have nilpotent centralizers are solvable.
- The F-T theorem tells us/is equivalent to the statement that odd-order groups whose non-identity elements have solvable centralizers are solvable.
That article also says that
In the odd order paper Feit and Thompson proceed by analyzing the structure of the centralizers inside a hypothetical minimal counterexample to their theorem.
In this blogpost, Terry Tao describes a "ladder" to the CFSG, with rungs in particular going from CA theorem (to the CN theorem) to the F-T theorem.
In summary: going up this ladder we are considering groups with (progressively less and less) nice properties regarding the centralizers of elements and still trying to prove that it is solvable (resulting in the CA, CN, and F-T theorems); and in the separate (but still related of course) subject of finding new simple groups, as Borcherds said we again focus on centralizers (this time of just involutions).
So it appears that at many different stages of proving the CFSG --- from the very "beginning" in "1955 The Brauer–Fowler theorem implies that the number of finite simple groups with given centralizer of involution is finite, suggesting an attack on the classification using centralizers of involutions.", to the late stage of discovering the Monster group --- people relied on this philosophy/idea of focusing on/studying centralizers.
Question: why is this? Why should having some information on what elements commute with each other in a group, and/or properties about that set of commuting elements (e.g. that it is abelian, nilpotent, etc.), tell you things about the group as a whole, and moreover give classification results about whole families of groups?
P.S. in the above-linked Terry blogpost on Brauer-Fowler, Terry does say
...the actual proof of the CFSG does not quite proceed along these lines. However, a significant portion of the argument is based on a generalisation of this strategy, in which the concept of a centraliser of an involution is replaced by the more general notion of a normaliser of a {p}-group, and one studies not just a single normaliser but rather the entire family of such normalisers and how they interact with each other (and in particular, which normalisers of {p}-groups commute with each other), motivated in part by the theory of Tits buildings for Lie groups...
So perhaps I am giving the centralizer-strategy a little too much credit. But Terry's blogpost on the Frobenius and Suzuki theorems linked this very detailed (and a bit technical) survey by Solomon, and Ctrl-F "CG" finds occurences of centralizers $C_G(\bullet)$ all over the place (along with of course other fancier subgroups: Fitting subgroup, Thompson subgroup, maximal normal semisimple subgroups, generalized Fitting subgroups, p-layer subgroups, etc.). Regardless, I think the main body of my post makes it clear that the centralizer-strategy was quite important and pursued/utilized in many different areas, and I would appreciate any insight as to why.