doubly transitive permutation representation
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A group G acts 2-transitively on a set S if it
acts The Acts of the Apostles (, ''Práxeis Apostólōn''; ) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message to the Roman Empire. Acts and the Gospel of Luke make up a two-par ...
transitively on the set of distinct ordered pairs \. That is, assuming (without a real loss of generality) that G acts on the left of S, for each pair of pairs (x,y),(w,z)\in S\times S with x \neq y and w\neq z, there exists a g\in G such that g(x,y) = (w,z). The group action is sharply 2-transitive if such g\in G is unique. A 2-transitive group is a group such that there exists a group action that's 2-transitive and faithful. Similarly we can define sharply 2-transitive group. Equivalently, gx = w and gy = z, since the induced action on the distinct set of pairs is g(x,y) = (gx,gy). The definition works in general with ''k'' replacing 2. Such multiply transitive permutation groups can be defined for any natural number ''k''. Specifically, a permutation group ''G'' acting on ''n'' points is ''k''-transitive if, given two sets of points ''a''1, ... ''a''''k'' and ''b''1, ... ''b''''k'' with the property that all the ''a''''i'' are distinct and all the ''b''''i'' are distinct, there is a group element ''g'' in ''G'' which maps ''a''''i'' to ''b''''i'' for each ''i'' between 1 and ''k''. The
Mathieu groups In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objec ...
are important examples.


Examples

Every group is trivially 1-transitive, by its action on itself by left-multiplication. Let S_n be the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
acting on \, then the action is sharply n-transitive. The group of n-dimensional similarities acts 2-transitively on \mathbb^n. In the case n=1 this action is sharply 2-transitive, but for n>1 it is not. The group of n-dimensional projective transforms ''almost'' acts sharply (n+2)-transitively on the n-dimensional
real projective space In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in the real space It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properti ...
\mathbb^n. The ''almost'' is because the (n+2) points must be in
general linear position In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that a ...
. In other words, the n-dimensional projective transforms act transitively on the space of projective frames of \mathbb^n.


Classifications of 2-transitive groups

Every 2-transitive group is a primitive group, but not conversely. Every
Zassenhaus group In mathematics, a Zassenhaus group, named after Hans Zassenhaus, is a certain sort of doubly transitive permutation group very closely related to rank-1 groups of Lie type. Definition A Zassenhaus group is a permutation group ''G'' on a finit ...
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 In mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating gro ...
and are all
almost simple group In mathematics, a group (mathematics), group is said to be almost simple if it contains a non-abelian group, abelian simple group and is contained within the automorphism group of that simple group – that is, if it fits between a (non-abelian) sim ...
s.


See also

* Multiply transitive group


References

* * * * * {{Citation , last1=Johnson , first1=Norman L. , last2=Jha , first2=Vikram , last3=Biliotti , first3=Mauro , title=Handbook of finite translation planes , publisher=Chapman & Hall/CRC , location=Boca Raton , series=Pure and Applied Mathematics , isbn=978-1-58488-605-1 , mr=2290291 , year=2007 , volume=289 Group actions