Mathieu group

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:

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.

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_{23}

and be done with it.

Video 1.

Mathieu group section of Why Do Sporadic Groups Exist? by Another Roof (2023)

Source. Only discusses Mathieu group but is very good at that.

k-transitive group

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TODO why do we care about this?

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

Classification of k-transitive groups

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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

2-transitive group

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Classification of 2-transitive groups

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Classification of 3-transitive groups

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

Classification of 4-transitive groups

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

Classification of 5-transitive groups

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Apparently only Mathieu group

M_{12}

and Mathieu group

M_{24}

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?