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Mathieu group

| 🗖 nosplit | ↑ parent "Sporadic group" | words: 239 | descendant words: 427 | descendants: 13
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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!
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Each is a permutation group on elements. There isn't an obvious algorithmic relationship between and the actual group.
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TODO initial motivation? Why did Mathieu care about k-transitive groups?
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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 and be done with it.
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Ancestors

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