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( R e d i r e c t e d f r o m D o u b l y t r a n s i t i v e p e r m u t a t i o n g r o u p )
A group
G
{\displaystyle G}
acts 2-transitively on a set
S
{\displaystyle S}
if it acts transitively on the set of distinct ordered pairs
{
(
x
,
y
)
∈
S
×
S
:
x
≠
y
}
{\displaystyle \{(x,y)\in S\times S:x\neq y\}}
. That is, assuming (without a real loss of generality) that
G
{\displaystyle G}
acts on the left of
S
{\displaystyle S}
, for each pair of pairs
(
x
,
y
)
,
(
w
,
z
)
∈
S
×
S
{\displaystyle (x,y),(w,z)\in S\times S}
with
x
≠
y
{\displaystyle x\neq y}
and
w
≠
z
{\displaystyle w\neq z}
, there exists a
g
∈
G
{\displaystyle g\in G}
such that
g
(
x
,
y
)
=
(
w
,
z
)
{\displaystyle g(x,y)=(w,z)}
.
The group action is sharply 2 -transitive if such
g
∈
G
{\displaystyle g\in G}
is unique.
A 2-transitive group is a group such that there exists a group action that's 2-transitive and faithful. Similarly we can define sharply 2 -transitive group .
Equivalently,
g
x
=
w
{\displaystyle gx=w}
and
g
y
=
z
{\displaystyle gy=z}
, since the induced action on the distinct set of pairs is
g
(
x
,
y
)
=
(
g
x
,
g
y
)
{\displaystyle g(x,y)=(gx,gy)}
.
The definition works in general with k replacing 2. Such multiply transitive permutation groups can be defined for any natural number k . Specifically, a permutation group G acting on n points is k -transitive if, given two sets of points a 1 , ... a k and b 1 , ... b k with the property that all the a i are distinct and all the b i are distinct , there is a group element g in G which maps a i to b i for each i between 1 and k . The Mathieu groups are important examples.
Examples [ edit ]
Every group is trivially 1-transitive, by its action on itself by left-multiplication.
Let
S
n
{\displaystyle S_{n}}
be the symmetric group acting on
{
1
,
.
.
.
,
n
}
{\displaystyle \{1,...,n\}}
, then the action is sharply n-transitive.
The group of n-dimensional homothety-translations acts 2-transitively on
R
n
{\displaystyle \mathbb {R} ^{n}}
.
The group of n-dimensional projective transforms almost acts sharply (n+2)-transitively on the n-dimensional real projective space
R
P
n
{\displaystyle \mathbb {RP} ^{n}}
. The almost is because the (n+2) points must be in general linear position . In other words, the n-dimensional projective transforms act transitively on the space of projective frames of
R
P
n
{\displaystyle \mathbb {RP} ^{n}}
.
Classifications of 2-transitive groups [ edit ]
Every 2-transitive group is a primitive group , but not conversely. Every Zassenhaus group is 2-transitive, but not conversely. The solvable 2-transitive groups were classified by Bertram Huppert and are described in the list of transitive finite linear groups . The insoluble groups were classified by (Hering 1985 ) using the classification of finite simple groups and are all almost simple groups .
See also [ edit ]
References [ edit ]
Dixon, John D.; Mortimer, Brian (1996), Permutation groups , Graduate Texts in Mathematics, vol. 163, Berlin, New York: Springer-Verlag , ISBN 978-0-387-94599-6 , MR 1409812
Hering, Christoph (1985), "Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II", Journal of Algebra , 93 (1 ): 151–164, doi :10.1016/0021-8693(85 )90179-6 , ISSN 0021-8693 , MR 0780488
Huppert, Bertram (1957), "Zweifach transitive, auflösbare Permutationsgruppen", Mathematische Zeitschrift , 68 : 126–150, doi :10.1007/BF01160336 , ISSN 0025-5874 , MR 0094386
Huppert, Bertram; Blackburn, Norman (1982), Finite groups. III. , Grundlehren der Mathematischen Wissenschaften, vol. 243, Berlin-New York: Springer-Verlag, ISBN 3-540-10633-2 , MR 0650245
Johnson, Norman L.; Jha, Vikram; Biliotti, Mauro (2007), Handbook of finite translation planes , Pure and Applied Mathematics, vol. 289, Boca Raton: Chapman & Hall/CRC, ISBN 978-1-58488-605-1 , MR 2290291
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Multiply_transitive_group_action&oldid=1227185466 "
C a t e g o r y :
● P e r m u t a t i o n g r o u p s
H i d d e n c a t e g o r i e s :
● A r t i c l e s w i t h s h o r t d e s c r i p t i o n
● S h o r t d e s c r i p t i o n i s d i f f e r e n t f r o m W i k i d a t a
● T h i s p a g e w a s l a s t e d i t e d o n 4 J u n e 2 0 2 4 , a t 0 7 : 1 6 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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