Namespaces
Variants
Actions

Difference between revisions of "Transitive group"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (Undo revision 49016 by Ulf Rehmann (talk))
Tag: Undo
m (tex encoded by computer)
 
Line 1: Line 1:
A [[Permutation group|permutation group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t0938001.png" /> such that each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t0938002.png" /> can be taken to any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t0938003.png" /> by a suitable element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t0938004.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t0938005.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t0938006.png" /> is the unique [[Orbit|orbit]] of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t0938007.png" />. If the number of orbits is greater than 1, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t0938008.png" /> is said to be intransitive. The orbits of an intransitive group are sometimes called its domains of transitivity. For an intransitive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t0938009.png" /> with orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380010.png" />,
+
<!--
 +
t0938001.png
 +
$#A+1 = 84 n = 0
 +
$#C+1 = 84 : ~/encyclopedia/old_files/data/T093/T.0903800 Transitive group
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380011.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
and the restriction of the group action to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380012.png" /> is transitive.
+
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} $,
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380013.png" /> be a subgroup of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380014.png" /> and let
+
$$
 +
= X _ {1} \cup \dots \cup X _ {s} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380015.png" /></td> </tr></table>
+
and the restriction of the group action to  $  X _ {i} $
 +
is transitive.
  
be the decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380016.png" /> into right cosets with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380017.png" />. Further, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380018.png" />. Then the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380019.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380020.png" />. This action is transitive and, conversely, every transitive action is of the above type for a suitable subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380022.png" />.
+
Let  $  H $
 +
be a subgroup of a group  $  G $
 +
and let
  
An action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380023.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380025.png" />-transitive, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380026.png" />, if for any two ordered sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380027.png" /> distinct elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380030.png" />, there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380032.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380033.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380034.png" /> possesses just one anti-reflexive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380035.png" />-orbit. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380036.png" />, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380038.png" />-transitive group is called multiply transitive. An example of a doubly-transitive group is the group of affine transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380040.png" />, of some field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380041.png" />. Examples of triply-transitive groups are the groups of fractional-linear transformations of the projective line over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380042.png" />, that is, transformations of the form
+
$$
 +
= H \cup Hx _ {1} \cup \dots \cup Hx _ {s - 1 }
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380043.png" /></td> </tr></table>
+
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
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380044.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm det}  \left \|
  
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380045.png" />-transitive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380046.png" /> is said to be strictly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380048.png" />-transitive if only the identity permutation can leave <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380049.png" /> distinct elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380050.png" /> fixed. The group of affine transformations and the group of fractional-linear transformations are examples of strictly doubly- and strictly triply-transitive groups.
+
\begin{array}{ll}
 +
a  & b  \\
 +
c  & d  \\
 +
\end{array}
 +
\right \|  \neq  0.
 +
$$
  
The finite [[Symmetric group|symmetric group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380051.png" /> (acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380052.png" />) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380053.png" />-transitive. The finite [[Alternating group|alternating group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380054.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380055.png" />-transitive. These two series of multiply-transitive groups are the obvious ones. Two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380056.png" />-transitive groups, namely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380058.png" />, are known, as well as two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380059.png" />-transitive groups, namely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380061.png" /> (see [[#References|[3]]] and also [[Mathieu group|Mathieu group]]). There is the conjecture that apart from these four groups there are no non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380062.png" />-transitive groups for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380063.png" />. 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.
+
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.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380064.png" />-Transitive groups have also been defined for fractional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380065.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380067.png" />. Namely, a permutation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380068.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380069.png" />-transitive if either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380070.png" />, or if all orbits of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380071.png" /> have the same length greater than 1. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380072.png" />, a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380073.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380074.png" />-transitive if the stabilizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380075.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380076.png" />-transitive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380077.png" /> (see [[#References|[3]]]).
+
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====
 
====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>
 
<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====
 
====Comments====
The degree of a permutation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380078.png" /> is the number of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380079.png" />. An (abstract) group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380080.png" /> is said to be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380082.png" />-transitive group if it can be realized as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380083.png" />-fold transitive permutation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380084.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380085.png" />-transitive permutation groups have been found. See the list and references in [[#References|[a1]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380086.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380087.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093800/t09380088.png" /> have been almost fully classified by use of the classification of finite simple groups [[#References|[a2]]].
+
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====
 
====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>
 
<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>

Latest revision as of 14:56, 7 June 2020


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
How to Cite This Entry:
Transitive group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transitive_group&oldid=49635
This article was adapted from an original article by L.A. Kaluzhnin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article