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 \|

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