
2-transitive group

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 formula_1 acts 2-transitively on a set formula_2 if it acts transitively on the set of distinct ordered pairs formula_3. That is, assuming (without a real loss of generality) that formula_1 acts on the left of formula_2, for each pair of pairs formula_6 with formula_7 and formula_8, there exists a formula_9 such that formula_10. Equivalently, formula_11 and formula_12, since the induced action on the distinct set of pairs is formula_13.
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 using the classification of finite simple groups and are all almost simple groups.