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  "body": "\n## Summary\n\nA group action is $k$-transitive when one group element can move any $k$ distinct points onto any other $k$ distinct points, in order.\nFor $k = 1$ this is a [[transitive action|02GK6HZYBT]], and larger $k$ means a stronger kind of symmetry.\nThe symmetric group $S_n$ is $n$-transitive and the alternating group $A_n$ is $(n-2)$-transitive on $n$ points, and a $2$-transitive group is always [[primitive|Y10S0751KG]].\nApart from symmetric and alternating groups, finite multiply transitive groups are rare: the [[Mathieu groups|T1KR4KPX8P]] are famous examples.\n\n## Explanations\n\n### L1 human\n\nA group acting on a set is $2$-transitive if, for any two ordered pairs of different points, some element sends the first pair to the second.\nThink of a set of seats and a group of allowed ways to reseat people.\nIf the group is $2$-transitive, any two people can be moved into any two chosen seats.\nAll permutations of $n$ points form an $n$-transitive group.\nThe rotations and reflections of a square act transitively on its four corners, but not $2$-transitively: two neighbouring corners cannot be moved onto two opposite corners.\n",
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