2-transitive group

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A 2-transitive group is a transitive group used in group theory in which the stabilizer subgroup of every point acts transitively on the remaining points. Equivalently, a group acts 2-transitively on a set if it acts transitively on the set of distinct ordered pairs . That is, assuming (without a real loss of generality) that acts on the left of , for each pair of pairs with and , there exists a such that . Equivalently, and , since the induced action on the distinct set of pairs is .

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Classifications of 2-transitive groups

Every 2-transitive group is a primitive group, but not conversely. Every Zassenhaus group is 2-transitive, but not conversely. The solvable 2-transitive groups were classified by Bertram Huppert and are described in the list of transitive finite linear groups. The insoluble groups were classified by ( Hering 1985 ) using the classification of finite simple groups and are all almost simple groups.

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