Transitive group

A permutation group such that each element can be taken to any element by a suitable element , that is, . In other words, is the unique orbit of the group . If the number of orbits is greater than 1, then is said to be intransitive. The orbits of an intransitive group are sometimes called its domains of transitivity. For an intransitive group with orbits ,

and the restriction of the group action to is transitive.

Let be a subgroup of a group and let

be the decomposition of into right cosets with respect to . Further, let . Then the action of is defined by . This action is transitive and, conversely, every transitive action is of the above type for a suitable subgroup of .

An action is said to be -transitive, , if for any two ordered sets of distinct elements and , , there exists an element such that for all . In other words, possesses just one anti-reflexive -orbit. For , a -transitive group is called multiply transitive. An example of a doubly-transitive group is the group of affine transformations , , of some field . Examples of triply-transitive groups are the groups of fractional-linear transformations of the projective line over a field , that is, transformations of the form

where

A -transitive group is said to be strictly -transitive if only the identity permutation can leave distinct elements of fixed. The group of affine transformations and the group of fractional-linear transformations are examples of strictly doubly- and strictly triply-transitive groups.

The finite symmetric group (acting on ) is -transitive. The finite alternating group is -transitive. These two series of multiply-transitive groups are the obvious ones. Two -transitive groups, namely and , are known, as well as two -transitive groups, namely and (see [3] and also Mathieu group). There is the conjecture that apart from these four groups there are no non-trivial -transitive groups for . This conjecture has been proved, using the classification of finite simple non-Abelian groups [6]. Furthermore, using the classification of the finite simple groups, the classification of multiply-transitive groups can be considered complete.

-Transitive groups have also been defined for fractional of the form , . Namely, a permutation group is said to be -transitive if either , or if all orbits of have the same length greater than 1. For , a group is -transitive if the stabilizer is -transitive on (see [3]).

References

[1]	C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)
[2]	P. Hall, "The theory of groups" , Macmillan (1959)
[3]	H. Wielandt, "Finite permutation groups" , Acad. Press (1968) (Translated from German)
[4]	D. Passman, "Permutation groups" , Benjamin (1968)
[5]	D.G. Higman, "Lecture on permutation representations" , Math. Inst. Univ. Giessen (1977)
[6]	P.J. Cameron, "Finite permutation groups and finite simple groups" Bull. London Math. Soc. , 13 (1981) pp. 1–22

Comments

The degree of a permutation group is the number of elements of . An (abstract) group is said to be a -transitive group if it can be realized as a -fold transitive permutation group .

Due to the classification of finite simple groups, all -transitive permutation groups have been found. See the list and references in [a1].

An important concept for transitive permutation groups is the permutation rank. It can be defined as the number of orbits of on .

Primitive permutation groups with permutation rank have been almost fully classified by use of the classification of finite simple groups [a2].

References

[a1]	A. Cohen, H. Zantema, "A computation concerning doubly transitive permutation groups" J. Reine Angew. Math. , 347 (1984) pp. 196–211
[a2]	A.E. Brouwer, A.M. Cohen, A. Neumaier, "Distance regular graphs" , Springer (1989) pp. 229