Transitive group

A permutation group $ ( G, X) $ such that each element $ x \in X $ can be taken to any element $ y \in X $ by a suitable element $ \gamma \in G $, that is, $ x ^ \gamma = y $. In other words, $ X $ is the unique orbit of the group $ ( G, X) $. If the number of orbits is greater than 1, then $ ( G, X) $ is said to be intransitive. The orbits of an intransitive group are sometimes called its domains of transitivity. For an intransitive group $ ( G, X) $ with orbits $ X _ {i} $,

$$ X = X _ {1} \cup \dots \cup X _ {s} , $$

and the restriction of the group action to $ X _ {i} $ is transitive.

Let $ H $ be a subgroup of a group $ G $ and let

$$ G = H \cup Hx _ {1} \cup \dots \cup Hx _ {s - 1 } $$

be the decomposition of $ G $ into right cosets with respect to $ H $. Further, let $ X = \{ Hx _ {i} \} $. Then the action of $ ( G, X) $ is defined by $ ( Hx _ {i} ) ^ {g} = Hx _ {i} g $. This action is transitive and, conversely, every transitive action is of the above type for a suitable subgroup $ H $ of $ G $.

An action $ ( G, X) $ is said to be $ k $- transitive, $ k \in \mathbf N $, if for any two ordered sets of $ k $ distinct elements $ ( x _ {1} \dots x _ {k} ) $ and $ ( y _ {1} \dots y _ {k} ) $, $ x _ {i} , y _ {i} \in X $, there exists an element $ \gamma \in G $ such that $ y _ {i} = x _ {i} ^ \gamma $ for all $ i = 1 \dots k $. In other words, $ ( G, X) $ possesses just one anti-reflexive $ k $- orbit. For $ k \geq 2 $, a $ k $- transitive group is called multiply transitive. An example of a doubly-transitive group is the group of affine transformations $ x \mapsto ax + b $, $ 0 \not\equiv a, b \in K $, of some field $ K $. Examples of triply-transitive groups are the groups of fractional-linear transformations of the projective line over a field $ K $, that is, transformations of the form

$$ x \mapsto \frac{ax + b }{cx + d } ,\ \ a, b, c, d, x \in K \cup \{ \infty \} , $$

where

$$ \mathop{\rm det} \left \| \begin{array}{ll} a & b \\ c & d \\ \end{array} \right \| \neq 0. $$

A $ k $- transitive group $ ( G, X) $ is said to be strictly $ k $- transitive if only the identity permutation can leave $ k $ distinct elements of $ X $ 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 $ S _ {n} $( acting on $ \{ 1 \dots n \} $) is $ n $- transitive. The finite alternating group $ A _ {n} $ is $ ( n - 2) $- transitive. These two series of multiply-transitive groups are the obvious ones. Two $ 4 $- transitive groups, namely $ M _ {11} $ and $ M _ {23} $, are known, as well as two $ 5 $- transitive groups, namely $ M _ {12} $ and $ M _ {24} $( see [3] and also Mathieu group). There is the conjecture that apart from these four groups there are no non-trivial $ k $- transitive groups for $ k \geq 4 $. 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.

$ k $- Transitive groups have also been defined for fractional $ k $ of the form $ m + 1/2 $, $ m = 0, 1 ,\dots $. Namely, a permutation group $ ( G, X) $ is said to be $ 1/2 $- transitive if either $ | X | = 1 $, or if all orbits of $ ( G, X) $ have the same length greater than 1. For $ n > 1 $, a group $ ( G, X) $ is $ ( n + 1/2) $- transitive if the stabilizer $ ( G, X) $ is $ ( n - 1/2) $- transitive on $ X $( 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 $ ( G, X) $ is the number of elements of $ X $. An (abstract) group $ G $ is said to be a $ k $- transitive group if it can be realized as a $ k $- fold transitive permutation group $ ( G, X) $.

Due to the classification of finite simple groups, all $ 2 $- 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 $ G $ on $ X \times X $.

Primitive permutation groups with permutation rank $ \leq 3 $ 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