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A [[Permutation group|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|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|symmetric group]]  $  S _ {n} $(
acting on  $  \{ 1 \dots n \} $) 
is  $  n $-
transitive. The finite [[Alternating group|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 [[#References|[3]]] and also [[Mathieu group|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 [[#References|[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 [[#References|[3]]]).

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

====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 [[#References|[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 [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Cohen,   H. Zantema,   "A computation concerning doubly transitive permutation groups"  ''J. Reine Angew. Math.'' , '''347'''  (1984)  pp. 196–211</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.E. Brouwer,   A.M. Cohen,   A. Neumaier,   "Distance regular graphs" , Springer  (1989)  pp. 229</TD></TR></table>