As shown in Video "Simple Groups - Abstract Algebra by Socratica (2018)", this can be split up into two steps:This split is sometimes called the "Jordan-Hölder program" in reference to the authors of the jordan-Holder Theorem.
Good lists to start playing with:
History: math.stackexchange.com/questions/1587387/historical-notes-on-the-jordan-h%C3%B6lder-program
It is generally believed that no such classification is possible in general beyond the simple groups.
This dude has done well.
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:
- cyclic groups: infinitely many, one for each prime order. Non-prime orders are not simple. These are the only Abelian ones.
- alternating groups of order 4 or greater: infinitely many
- groups of Lie type: a contains several infinite families
- sporadic groups: 26 or 27 of them depending on definitions
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:
- Chevalley groups
- TODO the rest
The first in this family discovered were a subset of the Chevalley groups by Galois: , 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 , 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 by Galois. You should understand that one first.
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 is a permutation group on elements. There isn't an obvious algorithmic relationship between 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 and be done with it.
- 22 is 3-transitive but not 4-transitive.
- four of them (11, 12, 23 and 24) are the only sporadic 4-transitive groups as per the classification of 4-transitive groups (no known simpler proof as of 2021), which sounds like a reasonable characterization. Note that 12 and 25 are also 5 transitive.
Mathieu group section of Why Do Sporadic Groups Exist? by Another Roof (2023)
Source. Only discusses Mathieu group but is very good at that.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 and Mathieu group .
www.maths.qmul.ac.uk/~pjc/pps/pps9.pdf mentions:Hmm, is that 54, or more likely 5 and 4?
The automorphism group of the extended Golay code is the 54-transitive Mathieu group . This is one of only two finite 5-transitive groups other than symmetric and alternating groups
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 .
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
Group theory, abstraction, and the 196,883-dimensional monster by 3Blue1Brown (2020)
Source. Too basic, starts motivating groups themselves, therefore does not give anything new or rare.TODO clickbait, or is it that good?
Uniqueness results for the composition series of a group.
Besides the understandable Wikipedia definition, Video "Simple Groups - Abstract Algebra by Socratica (2018)" gives an understandable one:
Given a finite group and a simple group , find all groups such that is a normal subgroup of and .
We don't really know how to make up larger groups from smaller simple groups, which would complete the classification of finite groups:
In particular, this is hard because you can't just take the direct product of groups to retrieve the original group: Section "Relationship between the quotient group and direct products".
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