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 [math]\displaystyle{ G }[/math] acts 2-transitively on a set [math]\displaystyle{ S }[/math] if it acts transitively on the set of distinct ordered pairs [math]\displaystyle{ \{ (x,y) \in S \times S : x \neq y \} }[/math]. That is, assuming (without a real loss of generality) that [math]\displaystyle{ G }[/math] acts on the left of [math]\displaystyle{ S }[/math], for each pair of pairs [math]\displaystyle{ (x,y),(w,z)\in S\times S }[/math] with [math]\displaystyle{ x \neq y }[/math] and [math]\displaystyle{ w\neq z }[/math], there exists a [math]\displaystyle{ g\in G }[/math] such that [math]\displaystyle{ g(x,y) = (w,z) }[/math]. Equivalently, [math]\displaystyle{ gx = w }[/math] and [math]\displaystyle{ gy = z }[/math], since the induced action on the distinct set of pairs is [math]\displaystyle{ g(x,y) = (gx,gy) }[/math].

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.

See also

• Multiply transitive group

References

• Dixon, John D.; Mortimer, Brian (1996), Permutation groups, Graduate Texts in Mathematics, 163, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94599-6, https://archive.org/details/permutationgroup0000dixo 
• 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 
• Huppert, Bertram (1957), "Zweifach transitive, auflösbare Permutationsgruppen", Mathematische Zeitschrift 68: 126–150, doi:10.1007/BF01160336, ISSN 0025-5874 
• Huppert, Bertram; Blackburn, Norman (1982), Finite groups. III., Grundlehren der Mathematischen Wissenschaften, 243, Berlin-New York: Springer-Verlag, ISBN 3-540-10633-2 
• Johnson, Norman L.; Jha, Vikram; Biliotti, Mauro (2007), Handbook of finite translation planes, Pure and Applied Mathematics, 289, Boca Raton: Chapman & Hall/CRC, ISBN 978-1-58488-605-1