= Two-transitive group action
= Two-transitive action
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= Two-transitive
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= Doubly transitive group action
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= Two-transitivity
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A <group action> on a <set> $X$ with at least two points is two-transitive when it acts transitively on ordered pairs of distinct points. Equivalently, the <group action> is <transitive> and the <stabilizer subgroup> of one point acts transitively on the remaining points. First move the initial first point to the target first point, then use that <stabilizer subgroup> to move the second point. Conversely, two-transitivity supplies both kinds of moves.
The transitivity requirement cannot be omitted: a <group> that fixes one point and permutes all others has a <stabilizer subgroup> transitive on the remaining points, while the whole <group action> is not even transitive. This is the $n=2$ case of an <n-transitive group action>.
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