= Mathieu group
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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 group>[sporadic] <k-transitive group>[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 (group)> of $M_23$ and be done with it.
\Video[https://youtu.be/dxRf3vHbuoA?t=603]
{title=<Mathieu group> section of Why Do Sporadic Groups Exist? by Another Roof (2023)}
{description=Only discusses <Mathieu group> but is very good at that.}