Classification of finite simple groups

Ciro Santilli is very fond of this result: the beauty of mathematics.

How can so much complexity come out from so few rules?

How can the proof be so long (thousands of papers)?? Surprise!!

And to top if all off, the awesomely named monster group could have a relationship with string theory via the monstrous moonshine?

all science is either physics or stamp collecting comes to mind.

The classification contains:

In the classification of finite simple groups, groups of Lie type are a set of infinite families of simple lie groups. These are the other infinite families besides te cyclic groups and alternating groups.

A decent list at: en.wikipedia.org/wiki/List_of_finite_simple_groups, en.wikipedia.org/wiki/Group_of_Lie_type is just too unclear. The groups of Lie type can be subdivided into:

The first in this family discovered were a subset of the Chevalley groups $A_n(q)$ by Galois: $PSL(2,p)$, so it might be a good first one to try and understand what it looks like.

TODO understand intuitively why they are called of Lie type. Their names $A_n$, $B_n$ seem to correspond to the members of the classification of simple Lie groups which are also named like that.

But they are of course related to Lie groups, and as suggested at Video "Yang-Mills 1 by David Metzler (2011)" part 2, the continuity actually simplifies things.

They are the finite projective special linear groups.

This was the first infinite family of simple groups discovered after the simple cyclic groups and alternating groups. The first case discovered was $PSL(2,p)$ by Galois. You should understand that one first.

Examples of exceptional objects.

Contains the first sporadic groups discovered by far: 11 and 12 in 1861, and 22, 23 and 24 in 1973. And therefore presumably the simplest! The next sporadic ones discovered were the Janko groups, only in 1965!

Each $M_n$ is a permutation group on $n$ elements. There isn't an obvious algorithmic relationship between $n$ and the actual group.

TODO initial motivation? Why did Mathieu care about k-transitive groups?

Their; k-transitive group properties seem to be the main characterization, according to Wikipedia:Looking at the classification of k-transitive groups we see that the Mathieu groups are the only families of 4 and 5 transitive groups other than symmetric groups and alternating groups. 3-transitive is not as nice, so let's just say it is the stabilizer of $M_{2}3$ and be done with it.

TODO why do we care about this?

Note that if a group is k-transitive, then it is also k-1-transitive.

TODO this would give a better motivation for the Mathieu group

Higher transitivity: mathoverflow.net/questions/5993/highly-transitive-groups-without-assuming-the-classification-of-finite-simple-g

Might be a bit complex: math.stackexchange.com/questions/698327/classification-of-triply-transitive-finite-groups

en.wikipedia.org/w/index.php?title=Mathieu_group&oldid=1034060469#Multiply_transitive_groups is a nice characterization of 4 of the Mathieu groups.

Apparently only Mathieu group $M_{1}2$ and Mathieu group $M_{2}4$.

www.maths.qmul.ac.uk/~pjc/pps/pps9.pdf mentions:

The automorphism group of the extended Golay code is the 54-transitive Mathieu group $M_{24}$. This is one of only two finite 5-transitive groups other than symmetric and alternating groups

Hmm, is that 54, or more likely 5 and 4?

scite.ai/reports/4-homogeneous-groups-EAKY21 quotes link.springer.com/article/10.1007%2FBF01111290 which suggests that is is also another one of the Mathieu groups, math.stackexchange.com/questions/698327/classification-of-triply-transitive-finite-groups#comment7650505_3721840 and en.wikipedia.org/wiki/Mathieu_group_M12 mentions M_12.

math.stackexchange.com/questions/700235/is-there-an-easy-proof-for-the-classification-of-6-transitive-finite-groups says there aren't any non-boring ones.

A master thesis reviewing its results: scholarworks.sjsu.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&httpsredir=1&article=5051&context=etd_theses

TODO clickbait, or is it that good?

Uniqueness results for the composition series of a group.