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  "body": "\n## Definition\n\nLet a group $G$ [[act|0WCY42EV9V]] on a set $\\Omega$, and let $k$ be a positive integer with $k \\le \\abs{\\Omega}$.\nThe action is *$k$-transitive* if for all $k$-tuples $(x_1, \\dots, x_k)$ and $(y_1, \\dots, y_k)$ of pairwise distinct elements of $\\Omega$ there is $g \\in G$ with $g x_i = y_i$ for $i = 1, \\dots, k$.\nThe action is *sharply $k$-transitive* if, moreover, this element $g$ is unique for all such pairs of $k$-tuples.\n\n## Remarks\n\nA $1$-transitive action is a [[transitive action|02GK6HZYBT]].\nA $k$-transitive action is $j$-transitive for every $j \\le k$.\nFor $\\abs{\\Omega} \\ge 2$, an action is $2$-transitive if and only if it is transitive and the [[stabiliser|0AYMP41THN]] $G_\\omega$ of one point $\\omega$ is transitive on $\\Omega \\setminus \\set{\\omega}$.\n\nExample: the [[symmetric group|0515A6D2Y1]] $S_n$ is $n$-transitive on $\\set{1, \\dots, n}$, since any bijection between two orderings of distinct points extends to a permutation.\nFor $n \\ge 3$, the [[alternating group|0Z5905Q73D]] $A_n$ is $(n-2)$-transitive.\nIndeed, let $\\sigma \\in S_n$ do the job, and let $\\tau$ swap the two points not among $y_1, \\dots, y_{n-2}$.\nThen $\\tau \\sigma$ also does the job, and one of $\\sigma$ and $\\tau \\sigma$ is even.\nFor a [[finite field|0ZXZ82FTX2]] $\\F_q$, the maps $x \\mapsto ax + b$ with $a \\ne 0$ act sharply $2$-transitively on $\\F_q$.\n\nNon-example: the eight symmetries of a square, acting on its corners $1, 2, 3, 4$ in cyclic order.\nThe action is transitive, but no symmetry sends the pair $(1, 2)$ of neighbouring corners to the pair $(1, 3)$ of opposite corners, so it is not $2$-transitive.\n\nEach $2$-transitive action is [[primitive|Y10S0751KG]].\nCamille Jordan showed that for $k \\ge 4$ the only finite sharply $k$-transitive groups are $S_k$, $S_{k+1}$, $A_{k+2}$, and the [[Mathieu groups|T1KR4KPX8P]] $M_{11}$ for $k = 4$ and $M_{12}$ for $k = 5$.\n",
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