Multiply transitive group action

In group theory, a branch of mathematics, a 2-transitive group is a transitive permutation group in which the stabilizer subgroup of every point acts transitively on the remaining points. Equivalently, a permutation group acts 2-transitively on a set $S$ if it acts transitively on the set of distinct ordered pairs $\{(x,y)\in S\times S:x\neq y\}$ . 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.

References

Dixon, John D.; Mortimer, Brian (1996), Permutation groups, Graduate Texts in Mathematics, vol. 163, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94599-6, MR 1409812
Hering, Christoph (1985), "Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II", Journal of Algebra, 93 (1): 151–164, doi:10.1016/0021-8693(85)90179-6, ISSN 0021-8693, MR 0780488
Huppert, Bertram (1957), "Zweifach transitive, auflösbare Permutationsgruppen", Mathematische Zeitschrift, 68: 126–150, doi:10.1007/BF01160336, ISSN 0025-5874, MR 0094386
Huppert, Bertram; Blackburn, Norman (1982), Finite groups. III., Grundlehren der Mathematischen Wissenschaften, vol. 243, Berlin-New York: Springer-Verlag, ISBN 3-540-10633-2, MR 0650245
Johnson, Norman L.; Jha, Vikram; Biliotti, Mauro (2007), Handbook of finite translation planes, Pure and Applied Mathematics, vol. 289, Boca Raton: Chapman & Hall/CRC, ISBN 978-1-58488-605-1, MR 2290291